IN  MEMORIAM 
FLORIAN  CAJORl 


^^/cE^^  <:?^'>^ 


Digitized  by  the  Internet  Arciiive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/colawarithmeticOOcolarich 


SCHOOL   ARITHMETIC 


ADVANCED  BOOK 


BY 

JOHN  M.  COLAW,  A.M. 

ASSOCIATE  EDITOR  OF  THE  AMERICAN  MATHEMATICAL  MONTHLT, 
MONTEREY,  VA. 


AND 


J.  K.   ELLWOOD,  A.M. 

PRINCIPAL  OF  THE  COLFAX   PUBLIC  SCHOOL,   PITTSBURG,   PA.,  AUTHOR  OF 
TABLE  BOOK   AND   TEST  PROBLEMS    IN    ELEMENTARY  MATHEMATICS. 


t 


RICHM02TD,  VA. 

B.  F.  JOHNSON  PUBLISHING  CO. 

1900 


Copyright,  1900,  by 
J.  M.  COLAW  AND  J.  K.  ELL  WOOD 


CAJORl 


/.  •..:•  • 


PREFACE. 

To  meet  the  needs  of  progre8si\Le  teachers^in  the  best  pub- 
lic schools  has  been  the  first  aim  of  the  authors  in  writing 
the  present  work.  In  the.  effort  to  carry  out  this  purpose 
great  care  has  been  taken  to  make  the  book  modern  yet  con- 
servative. It  is  believed  that  an  examination  will  show  the 
book  to  be  sound  in  theory,  modern  in  method,  strong  in  in- 
ductive work,  clear  and  accurate  in  all  statements,  correct  in 
its  teaching  of  business  practice,  and  copious  and  varied  in 
its  supply  of  practical  work  and  problems.  Special  attention 
is  called  to  the  following  features  : 

1.  Inductive  Method.  The  inductive  method  is  applied 
throughout  the  major  part  of  the  book.  New  topics  are  in- 
troduced by  carefully  prepared  questions  and  suggestions 
designed  to  develop  in  the  pupiFs  mind  a  correct  understand- 
ing of  the  principles  to  be  taught,  and  to  give  clear  insight 
into  arithmetical  relations  and  processes. 

2.  Oral  and  Written  Work  are  given  with  every  appro- 
priate subject,  not  only  to  injpress  the  princii)les  thereof, 
but  also  to  provide  ample  practice  in  making  solutions  and 
abundant  material  for  mental  discipline.  To  lead  the  pupil 
to  think  in  general  terms,  as  well  as  to  familiarize  him  with 
the  use  of  symbols  and  lay  a  foundation  for  the  study  of  alge- 
bra, numerical  exercises  are  followed  by  others  involving  the 
same  processes  with  letters. 

3.  Rules  and  Definitions.  Few  rules  are  given,  and 
these  usually  in  the  form  of  directions.  At  the  same  time 
there  has  been  no  hesitancy  in  giving  a  few  rules  where  it 
was  thought  they  would  assist  the  pupil  in  gaining  a  correct 


4  PREFACE. 

and  intelligent  understanding  of  the  processes,  or  help  him 
to  neat  and  orderly  methods,  but  in  no  place  is  dependence 
on  rules  encouraged.  The  definitions  are  brief  but  accurate  ; 
unnecessary  ones  are  omitted. 

4.  Form  of  Solutions.  The  solutions  given  are  intended 
as  far  as  possible  to  suggest  the  reasons  for  the  various  steps 
and  to  render  unnecessary  the  lengthy,  cumbrous,  and  tedious 
explanations  and  analyses  often  given  in  arithmetical  text- 
books. 

5.  Solutions  of  Problems,  with  Examples  for  Prac- 
tice. This  chapter  affords  the  pupil  not  only  examples  of 
neatness  and  orderly  procedure  in  making  solutions,  but  also 
a  basis  for  the  recognition  of  similar  problems  and  an  aid  in 
devising  definite  processes  for  their  solution. 

6.  Practice  Work  and  Problems.  The  book  is  believed 
to  be  unsuri^assed  in  the  abundance,  variety,  and  excellent 
character  of  its  practice  work  and  problems,  yet  the  teacher 
may  find  it  advisable  at  times  to  supply  additional  work 
along  special  lines. 

7.  Arrangement  of  Topics.  The  topical  plan  of  treat- 
ment has  in  the  main  been  adhered  to  as  giving  the  best 
adaptation  to  a  wider  range  of  schools  ;  but  where  it  could  be 
advantageously  done,  much  valuable  information  relating  to 
subsequent  topics  has  been  introduced  in  the  examples  of  the 
earlier  sections.  In  the  Primary  book  the  order  of  topics  is 
not  preserved,  but  the  subject  matter  is  presented  in  the 
order  that  has  been  found  to  be  most  helpful,  most  stimulat- 
ing, and  most  practical,  the  arrangement  being  determined 
by  the  order  of  the  child's  mental  development  rather  than 
by  the  logic  of  the  subject  itself.  In  this  book  the  usual 
arrangement  of  topics  has  been  departed  from  in  a  few 
instances.     We  note  the  following  : 

(a)  Factoring  is  clearly  taught,  but  the  G.  C.  D.,  which 
is  now  rarely  met  with  in  practical  life  and  which  is  difficult 
for  beginners  to  understand,  is  presented  where  it  can  be 


PREFACE.  5 

given  in  review  at  a  time  when  the  pupil  is  better  able  to 
grasp  the  logical  reasoning  it  presents. 

(b)  U.  S.  Money,  easily  understood  by  young  pupils,  is 
used  as  an  introduction  to  decimal  fractions. 

(c)  Decimal  Fractions  are  treated  before  common  frac- 
tions, because  now  more  generally  employed,  and  because  the 
formal  treatment  in  connection  with  integers  is  more  easily 
understood,  the  notation  on  the  right  of  the  decimal  point 
being  as  easily  understood  as  that  on  the  left  if  the  decimal 
fractions  are  compared  with  integers  instead  of  with  common 
fractions.  In  the  Primary  book,  however,  the  common 
fraction  is  first  introduced,  because  the  first  fractions  with 
which  very  young  children  become  acquainted  are  the  half, 
quarter,  etc.,  and  because  these  are  more  simple  and  within 
the  range  of  the  child's  visualizing  power.  In  a  formal 
treatment,  however,  designed  for  older  pupils,  the  reasons 
are  reversed. 

(d)  Ratio  and  (Simple)  Proportion.  Ratio  is  given  in 
connection  with  Relation  of  Numbers  on  account  of  their 
intimate  connection.  Simple  Proportion,  furnishing  as  it 
does  a  good  basis  of  arithmetical  reasoning,  is  given  earlier 
than  usual  as  an  aid  in  improving  the  reasoning  ability  of 
the  pupil.  The  treatment  is  simple  and  not  burdened  with 
unnecessary  matter.  Compound  proportion  being  little  used 
is  given  in  the  Appendix  for  later  review. 

(e)  Practical  Mensuration  is  treated  in  connection  with 
square  and  cubic  measures,  thus  enriching  the  treatment, 
giving  greater  variety,  and  adding  interest  to  the  work  by 
showing  the  practical  uses  to  which  these  measures  are 
put. 

(f )  The  Metric  System  is  given  directly  after  Compound 
Numbers  for  purposes  of  comparison  as  well  as  to  provide  a 
variety  of  problems  in  the  supplementary  exercises  intended 
for  advanced  classes. 

(g)  Subjects  of  minor  importance  and  such  as  are  suited  to 


6  PREFACE. 

later  reviews  are  placed  in  the  Appendix,  while  subjects  of 
no  practical  value  in  a  modern  arithmetic  are  omitted. 

8.  Chapter  on  the  Equation.  The  importance  of  in- 
troducing this  simple  and  easy  chapter  cannot  well  be  over- 
estimated. Arithmetical  methods  are  adhered  to,  and  by- 
simple  inductive  exercises  and  comparative  solutions  the  con- 
ceptions of  the  use  of  letters  as  the. general  representatives  of 
positive  numbers,  and  of  the  equation  to  express  their  relations, 
are  developed  as  far  as  the  purpose  of  the  chapter  demands. 
By  the  aid  of  this  simple  chapter,  equations  which  have  a 
meaning  to  the  pupil  can  be  substituted  for  the  dead  for- 
mulas sometimes  used  in  percentage  problems  and  interest 
problems,  and  a  much  clearer  understanding  can  be  had  of 
such  subjects  as  the  greatest  common  divisor,  the  square  and 
cube  roots,  etc.  The  subject  is  further  developed  in  the  brief 
but  attractive  chapter  on  ^^  Introduction  to  Algebra"' given 
in  the  Appendix. 

9.  Division  of  Problems.  Experience  has  shown  that 
the  same  problems  should  not  be  solved  by  the  same  class 
from  term  to  term.  It  is  wiser  to  add  fresh  fuel  than  to  be 
constantly  stirring  the  old  coals.  The  pupil's  interest  must 
be  kept  alive.  With  tliis  in  view,  we  have  divided  the  prob- 
lems into  two  parts  ;  the  first  to  be  used  when  the  class  first 
goes  over  a  subject,  the  second  when  it  reviews  that  subject. 
For  a  similar  reason  the  reviews  have  also  been  so  divided. 

10.  Character  of  Problems.  In  the  preparation  of 
problems  tlie  actual  business  practice  of  to-day  has  been  kept 
in  mind.  In  addition  to  their  practical  and  matliematical 
value,  many  of  the  problems  furnish  mucli  useful  and  scien- 
tific information  that  is  reliable  and  strictly  up-to-date.  The 
aim  has  been  to  make  them  as  practical  and  «^5e/V<?  as  possible. 

11.  Scope  of  the  Work.  The  series  meets  the  needs 
not  only  of  the  primary  and  the  grammar  school,  but  also  of 
the  high  school.  For  the  latter  there  is  sufficient  theory,  and 
an  abundance  of  practice  work,  owing  to  the  ^'  Supplementary 


PREFACE.  7 

Exercises"  for  advanced  classes,  and  matter  included  in  the 
Appendix. 

12.  Bank  Practice  of  to-day  is  given  in  a  concise  and 
accurate  cliapter.  The  information  has  been  furnished  by 
officials  of  prominent  banks  in  the  States  mentioned  in  the 
exercises. 

13.  The  unique  treatment  of  various  processes  will  appeal 
to  the  intelligent  and  progressive  teacher.  The  merits  of 
these  processes  can  be  determined  only  by  careful  examination 
of  the  text  itself. 

14.  The  matter  and  method  of  the  Primary  book  are  de- 
signed to  serve  as  an  introduction  to  and  foundation  for  the 
more  formal  and  rigorous  treatment  presented  in  these  pages. 

The  authors  desire  to  express  their  thanks  for  valuable 
criticisms  and  suggestions  from  many  educators  during  the 
preparation  of  this  series.  They  especially  acknowledge 
their  indebtedness  to  Prof.  Frank  W.  Duke,  of  Hollins 
Institute,  Hollins,  Va.,  for  help  of  this  kind. 

J.  M.  COLAW, 
J.  K.  ELLWOOD. 
July  2,  1900. 


CONTENTS. 


PAGE 

Illustrations  and  Definitions— Units  and  Numbers       .        .  15 

Notation  and  Numeration  of  Integers 16 

The  Arabic  Notation 16 

United  States  Money 32 

Tlie  Roman  Notation 23 

Addition 25 

Subtraction 39 

Multiplication 52 

Division 64 

Review  Work 80 

Supplementary  Exercises 86 

Factors  and  Multiples 87 

Tests  of  Divisibility 88 

Factoring 88 

Least  Common  Multiple 90 

Cancellation 92 

United  States  Money 94 

Decimal  Fractions 100 

Reading  and  Writing  Decimals 102 

Addition  and  Subtraction 106 

Multiplication  and  Division 108 

Bills  and  Accounts 114 

Review  Work 117 

Supplementary  Exercises 120 

Common  Fractions 122 

Change  of  Form 124 

Addition 133 


10  CONTENTS. 

PAGE 

Supplementary  Exercises . 135 

Subtraction 136 

Supplementary  Exercises 138 

Multiplication 138 

Supplementary  Exercises 143 

Division 144 

Supplementary  Exercises 148 

Illustrative  Solutions,  with  Problems  for  Practice    .        .        .  150 

Relation  op  Numbers 156 

Aliquot  Parts 158 

Ratio 160 

Proportion      , 163 

Supplementary  Exercises 167 

The  Equation 169 

Supplementary  Exercises 177 

Review  Work 178 

Supplementary  Exercises 188 

Compound  Numbers 189 

Measures  of  Value 189 

Reduction 191 

Measures  of  Capacity 194 

Measures  of  Weight 196 

Measures  of  Extension 199 

Square  Measure 203 

Area    of    Parallelograms,      Triangles,     Trapezoids,     and 

Circles 205-310 

Supplementary  Exercises 213 

Measures  of  Volume 214 

Recta.igular  Solids,  Cube,  Volume 214 

Cubic  Measure 216 

Supplementary  Exercises 217 

Surveyors'  Measure 218 

Measures  of  Time 219 

The  Calendar 219 

Circular  or  Angular  Measure             221 

Addition .        .        .        .222 


CONTENTS.  11 

PAGE 

Subtraction 225 

Multiplication 227 

Division 228 

Review  Work 230 

Supplementary  Exercises 235 

Longitude  and  Time 236 

Standard  Time 240 

Date  Line 242 

Supplementary  Exercises 243 

Practical  Measurements 244 

Painting  and  Plastering 244 

Measurement  of  Lumber 245 

Brick  Work  and  Stone  Work 247 

Carpeting 248 

Papering 249 

Bins,  Cisterns,  etc 250 

The  Metric  System 252 

Supplementary  Exercises 258 

General  Review  Work 259 

Supplementary  Exercises 265 

Percentage 266 

Supplementary  Exercises 277 

Commercial  Discount 279 

Commission 282 

Profit  and  Loss 285 

Taxes 289 

State  and  Local  Taxes 290 

National  Taxes .         .291 

Review  Work 293 

Supplementary  Exercises ,         .  296 

Interest 298 

Promissory  Notes 302 

Partial  Payments 306 

Problems  in  Simple  Interest     .......  309 

Annual  Interest 312 

Compound  Interest 313 


12  '  CONTENTS. 

PAGE 

Banks  and  Bank  Discount 316" 

Making  Loans 317 

Discounting  Notes .         .         .  321 

Present  Worth  and  True  Discount 324 

Stocks  and  Bonds 325 

Review  Work 331 

Supplementary  Exercises 336 

Proportional  Parts 337 

Partnership 338 

General  Review  Work 341 

Supplementary  Exercises 353 

Powers  and  Roots 354 

Involution 354 

Evolution          .        .        .        .    • 357 

Square  Root 358 

Applications  of  Square  Root 362 

The  Right-Triangle,  Pythagorean  Theorem      .        .        .  363 

Cube  Root 365 

Applications  of  Cube  Root 369 

Miscellaneous  Problems 371 

Supplementary  Exercises 385 

APPENDIX. 

Mensuration 390 

Prisms 390 

The  Cylinder    . 392 

The  Cone  and  the  Pyramid 393 

The  Sphere 397 

Similar  Figures 398 

Supplementary  Problems 401 

Greatest  Common  Divisor 404 

Compound  Proportion 406 

Insurance 408 

Exchange 410 


CONTENTS.  13 


FAGE 

Average  of  Payments 415 

Casting  out  Nines 416 

Measures  of  Temperature 418 

Specific  Gravity 419 

Introduction  to  Algebra 421 


SCHOOL    ARITHMETIC. 


ILLUSTRATIONS   AND   DEFINITIONS. 

A  pail  contains  a  quantity  of  milk  which  we  wish  to 
measure.  We  take  a  quart  measure  and  till  it  six  times,  thus 
finding  how  much  milk  we  have. 

(a).  The  unit  of  measure  here  is  the  quart. 

(b).  The  number  of  times  we  filled  it  tells  us  how  many 
quarts  of  milk  there  are. 

(c).  Six  quarts  tells  us  two  things :  (1)  how  many- 
units  of  measure  the  pail  contains  ;  (2)  how  much  milk  is 
in  the  pail,  i.e.,  the  quantity  of  it. 

1.  A  Unit,  or  unit  of  measure,  is  any  quantity  by  com- 
parison with  which  any  other  quantity  of  the  same  kind 
is  measured. 

Thus,  when  we  buy  a  piece  of  cloth  containing  ten  yards,  one  yard 
is  the  unit.  When  we  measure  a  wall  with  a  ten-foot  pole,  the  unit 
is  ten  feet.  If  we  use  a  ruler  \  yard  long,  \  yard  is  the  unit.  If  a  box 
contains  6  dozen  pencils,  a  dozen  is  the  unit.  When  we  say  we  have  ten 
apples,  the  unit  (of  reference)  is  one  apple. 

2.  Number  denotes  how  many  units  make  a  measured 
quantity.  It  tells  how  many  times  the  unit  of  measure  is 
applied  or  repeated. 

Thus,  5,  3  pounds,  $8,  12  feet  are  numbers. 

A  number  always  answers  the  question,  "  How  many  ?  " 

3.  A  Pure,  or  Abstract,  number  is  one  that  tells  how 
many  times  the  unit  of  measure  is  applied  or  repeated.     It 


•"'•'•'.  :•.••• : .- . , :  /:.  .school  arithmetic. 

expresses  the  ratio  nf 

W«l  that  is,  the  rat  0  Z^lT'  '"  ""*^"-  °'  *"  ^«»e 
°»'t-  ^  ^'^''^  1"^"tity  to  its  measuring 

caned  co^t^rSr^Xr^'^;--"'"-"'"-- 
anything.  """''*•      They    denote   kow  much  of 

lor  the  sake  of  convenience.  ^'    ^^^^^garding  the  distinction 

^'  Arithmetic   treats  nf  , 
of-aberstobusinessandiiere    "'  ""'  ^'^'^  ^^^P«-«on 

^-f«fa^ror:;i:r::r  sr'?''  ^  ^'^»'~- 

-'^oie  .a  ,„a„tity_n.,  onH,  i  Jt;:;;!,- "-«-  is  the 

or  ^etters^rrr^:^ -ES^^^^^^    "-"^^  V  %ures         - 

8.  The  method  of  reading-  inf^o-r^  i 
%~  ietters  is  oaiied  .uirif-iri::^ J~'  ^^ 

Europe       The  symbols  of    his  „t  .         '"'''"'^"'^^'J  '■'  "'to 
onginated  among  the  Hin.l,         T^"""'  ''^^pt   the   zero 
ago.     The  zero  did  not  '1    '"  '"^'''  "'"''  «>«»  3,000  yeaTs' 
--:     The  European:,  a  S:?'  ''^""^  "^«  «'!•  cL'tr^ 
donng  the  twelfth  century  "««'^J^tem  from  the  ArabI 


NOTATION  AND  NUMERATION.  17 

10.  In  this  system  of  notation  ten  figures,  or  digits,  are 
used  to  express  numbers,  viz.: 

01234567         89 
zero  one  two  three  four  five  six  seven  eight  nine 
The  first  figure,  zero,  is  also  called  naught  and  cipher,  and 
signifies  no7ie.     It  is  used  with  other  figures  to  express  num- 
bers larger  than  9. 

Figures  are  not  numbers  ;  they  are  characters  or  symbols  used  to 
express  or  represent  numbers. 

ONES,  TENS,  AND  HUNDREDS. 

11.  In  counting  up  to  nine  we  count  by  ones.  After  9  we 
say  ten,  but  we  have  no  single  figure  to  express  this  number. 
Hence  to  express  ten  ones,  or  1  ten,  we  combine  zero  with  1, 
and  write  the  number  thus  :  10. 

12.  Principle. —  When  a  figure  stands  alone,  it  expresses 
ones.  When  two  figures  stand  side  hy  side,  the  one  on  the 
right  expresses  ones,  the  other  tens. 

Thus,  in  2\  the  4  represents  4  ones,  and  the  2  represents  2  tens. 
3  tens,  or  20,  is  called  tiventy. 

3  ''      ''  30,  ''      ''      thirty. 

4  ..      ..  40,  "      "      forty. 

5  "      "  50,  "      "      fifty. 

6  ''      ''  60,  ''      "      sixty. 

7  ''      ''  70,  ''      ''      seventy. 

8  ''      ''  80,  ''      ''      eighty. 

9  ''      "  90,  ''      "      ninety. 

13.  In  counting  more  than  nine  we  count  by  tens,  as 
above,  or  by  tens  and  ones,  as  follows  : 

1  ten  and  1  one,    or  11,  is  called  eleven. 
1  ten    "     2  ones,  or  12,  ''      '^      twelve. 
1  ten    ''    3  ones,  or  13,  "      "      thirteen. 
1  ten    "     4  ones,  or  14,  "      ''      fourteen. 
1  \Qn    ^'     5  ones,  or  15,  "      "      fifteen. 
2 


18  SCHOOL  ARITHMETIC. 

1  ten  and  6  ones,  or  16,  is  called  sixteen. 


1  ten    "     7  ones,  or  17,  '' 

(< 

seventeen. 

1  ten   ''     8  ones,  or  18,  '' 

(( 

eighteen. 

1  ten   ''     9  ones,  or  19,  '' 

it 

nineteen. 

2  tens,                     or  20,  '' 

a 

twenty. 

2  tens  ''     1  one,    or  21,  '' 

a 

twenty-one. 

3  tens  ''     2  ones,  or  32,  " 

(( 

thirty-two. 

4  tens  ''    3  ones,  or  43,  '' 

i( 

forty-three. 

8  tens  ''     7  ones,  or  87,  " 

<{ 

eighty-seven 

9  tens  ''     9  ones,  or  99,  " 

a 

ninety-nine. 

14.  What  does  each  figure  in  the 

following  express 

p 

1.  15            5.  38              9.  6L 

13. 

84            17. 

92 

2.  19            6.  43            10.  68 

14.  88            18. 

97 

3.  21            7.  47            11.  73 

15. 

79            19. 

90 

4.  26            8.  54            12.  59 

16. 

81             20. 

99 

16.  When  we  count  one  more  than  99,  we  have  nine  tens 
and  10  ones,  or  10  tens,  which  is  called  one  hundred.  To  ex- 
press this  the  figure  1  is  written  at  the  left  of  two  ciphers, 
thus  :  100. 

16.  Peinciple. —  When  three  figures  are  written  side  hy 
side,  the  one  at  the  left  expresses  hundreds. 

Thus,  in  324  the  4  represents  4  ones,  the  2  represents  2  tens,  and  the 
3  represents  3  hundreds. 

17.  A  number  expressed  by  three  figures  is  read  without 
the  word  and. 

Thus,  324  is  read  three  hundred  twenty-four. 

18.  Read  the  following  : 


17 

24 

48 

27 

32 

39 

41 

46 

29 

50 

53 

66 

65 

77 

84 

90 

72 

124 

231 

132 

213 

346 

427 

536 

175 

381 

450 

619 

680 

700 

798 

800 

870 

660 

739 

808 

987 

711 

444 

330 

602 

101 

507 

600 

789 

Write  the  third  vertical  column  in  words. 


NOTATION  AND  NUMERATION.  19 

19.  Express  the  following  by  figures  : 

1.  Fifty.  14.  One  hundred  fifty. 

2.  Forty-two.  15.  Two  hundred  fifty-four. 

3.  Sixty-nine.  16.  Three  hundred  seven. 

4.  Seventy-six.  17.  Six  hundred  forty-five. 

5.  Thirty-seven.  18.  Eight  hundred  sixteen. 

6.  Ninety-five.  19.  Four  hundred  fifty-six. 

7.  Eighty-eight.  20.  Five  hundred  twenty. 

8.  Sixty-seven.  21.  Five  hundred  nine. 

9.  Twenty-four.  22.  Seven  hundred  eighteen.. 

10.  Ninety-nine.  23.  Nine  hundred  three. 

11.  Thirty-eight.  24.  Six  hundred  sixty-six. 

12.  Seventy-three  26.  Eight  hundred  eighty. 

13.  Eighty-seven.  26.  Three  hundred  three. 

27.  Nine  hundreds,  three  tens,  seven  ones. 

28.  Seven  hundreds,  eight  ones,  nine  tens. 

29.  Three  ones,  seven  tens,  four  hundreds. 

30.  Write  from  dictation  the  numbers  in  Art.  18. 

Queries. — 1.  Wlien  1  stands  alone,  how  many  ones  does  it  express  ? 
How  many  when  it  stands  at  the  left  of  a  cipher  ?  At  the  left  of  two 
ciphers  ? 

2.  In  the  number  11,  which  1  expresses  the  greater  value  ?  How 
many  times  as  great  ?    In  the  number  111  ? 

20.  Principle. — A 7iy  figure  represents  ten  times  the  value 
it  would  represent  in  the  next  place  to  the  right,  or  one  tenth 
of  the  value  it  would  represent  m  the  next  place  to  the  left. 

21.  The  system  of  counting  by  tens  is  called  the  Decimal 
System.  In  the  Arabic  notation  ten  of  any  place  or  order 
make  one  of  the  next  higher  order,  hence  the  system  is  a 
decimal  one.  This  system  derives  its  greatest  importance 
from  the  use  of  zero,  which  renders  possible  the  giving  of 
place  value  to  the  figures. 

22.  The  ones  of  a  number  are  called  units  of  the  first 
order,  or  simply  imits;  the  tens  are  called  units  of  the  second 


20  SCHOOL  ARITHMETIC. 

order;  the  hundreds,  are  called  iinits  of  the  third  order,  and 
so  on. 

23.  For  convenience  in  writing  and  reading  numbers  the 
figures  are  divided  into  groups  of  three  figures  each,  begin- 
ning at  the  riglit.  Each  group  is  called  ii  period,  and  con- 
tains ones,  tens,  and  hundreds  of  that  period.  The  right- 
hand  group  is  called  the  period  of  units  ;  the  second  group, 
the  period  of  thousands  ;  the  third  group,  the  period  of  mil- 
lions, and  so  on. 

24.  The  system  of  notation  is  shown  in  the  following 

TABLE  : 


CO 

Cfl 

CO 

1 

CO 

C 

to  to 

. 

^  t3 

r;^  to 

^ 

r^  CO 

o^ 

/— V 

Orders.       -v  o 

15§ 

.„    CO 
O)  ^    CO 

ed-m 
illion 

IS 

ed-th 
ousai 
mds 

i2 

CO         -^ 

1  ts 

^X5;2 

±!  "S  f^ 

rS    £    O 

^^1 

|§§ 

^    Oi-G 

g    CO    to 

W^H 

WrHpq 

WHS 

Khh 

ffiHO 

2  0  5 

6  1  9 

0  7  0 

8  40 

5  3  2 

^"stiT 

^TtiT 

^T 

^sT' 

1st 

Period. 

Period. 

Period. 

Period. 

Period. 

Trillions. 

Billions. 

Millions. 

Thousands. 

Units. 

The  number  in  the  table  is  read  two  hundred  five  trillion, 
six  hundred  nineteen  billion,  seventy  million,  eight  hundred 
forty  thousand,  five  hundred  thirty-two. 

1.  The  left-hand  period  may  have  only  one  or  two  figures  ;  the  others 
must  contain  three  figures. 

2.  In  writing  large  numbers,  the  periods  may  be  separated  by  com- 
mas or  written  slightly  apart,  as  an  aid  in  reading  them. 

3.  Each  period  is  read  as  if  it  stood  alone,  the  name  of  the  period 
being  added  except  in  the  case  of  the  units'  period. 

4.  The  names  of  periods  above  trillions  are,  in  order,  quadrillions, 
quintillions,  sextillions,  septillions,  octillions,  nonillions,  decillions,  etc. 


Notation  and  numeration. 


21 


25. 

1. 
2. 
3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
lines. 


EXERCISES    IN    NUMERATION. 

Read  the  following  numbers  : 


2,375, 
5,732, 

8,360, 
4,07G, 

8,007, 


52374, 

48106, 

90245, 

13004, 

86730, 
71,248,359,024, 
42,095,384,217, 
14,703,692,581, 
60,482,604,826, 
90,007,080,500, 
1,928,374,650,437,689, 
1,234,567,891,098,765, 


327195, 

245680, 

503276, 

100538, 

736451, 
370,689,421, 
124,986,073, 
581,470,369, 
726,934,851, 
900,604,003, 


6,382,541,  2  906  738. 
7,328,911,  1  540  006. 
5,374,802,  8  000  325. 
2,704,190,  6  030  008. 
1,398,453,   5  111  290. 

4,210,893,657. 

7,563,980,124. 

2,581,470,369. 

8,276,354,090. 

3,415,628,072. 
7,129,384,756,140,632,687. 
3,470,153,647,005,296,304. 


Write  in  words  the  numbers  in  the  1st,  10th,  and  12th 


EXERCISES   IN    NOTATION. 


26.  The  periods  are  written  in  regular  order,  beginning 
with  the  highest.  If  any  order  of  units  of  a  period  is  lack- 
ing, a  cipher  must  be  put  in  its  place  ;  and  if  any  entire 
period  is  lacking,  its  three  places  must  be  filled  with  ciphers. 

27.  Write  the  following  in  figures  : 

1.  Three  thousand,  eight  hundred  forty-five. 

2.  Eighty-three  thousand,  seven  hundred  forty. 

3.  Seventy-two  thousand,  three  hundred  five. 

4.  Thirty-seven  thousand,  five  hundred  twenty-one. 

5.  Ninety  thousand,  ninety.     Four  thousand,  six. 

6.  Fifty  thousand,  five.     Seventeen  thousand,  one. 

7.  Ten  thousand.     Six  thousand,  one  hundred  nine. 

8.  Three  hundred  forty-two  thousand,  six  hundred  sev- 
enty-five. 

9.  One  hundred  six  thousand,  four. 

10.  Three  hundred  thousand,  three  hundred. 

11.  Eight  hundred  six  thousand,  seventy-five. 


22  SCHOOL  ARITHMETIC. 

12.  One  hundred  seventeen  thousand,  four  hundred  two. 

13.  Eight  million,  eight  thousand,  eight. 

14.  Fifteen  million,  fifteen  thousand,  thirty. 

15.  Five  hundred  two  million,  three  hundred  four. 

16.  Two  million,  six  thousand,  ninety-eight. 

17.  Six  hundred  twelve  thousand,  four  hundred  sixty-two. 

18.  Five  hundred  sixty  thousand,  four  hundred  fifty-six. 

19.  Ten  million,  one  thousand,  one  hundred. 

20.  Sixty-five  million,  one  hundred  eight  thousand. 

21.  Nine  hundred  eighty-seven  million,  four. 

22.  Four  hundred  thousand,  fifty-six. 

23.  One  hundred  ten  thousand,  ninety. 

24.  Nine  hundred  million,  three  hundred  eighty-one. 

25.  Six  million,  six  hundred  thousand,  sixty. 

26.  Fifteen  million,  nine  thousand,  seventeen. 

27.  Four  thousand,  three  hundred  six. 

28.  Fifty-four  thousand,  three  hundred  fifteen. 

29.  Five  hundred  seven  million,  sixty-thousand,  two. 

30.  Nine  billion,  four  hundred  million,  eighty. 

31.  Write  from  dictation  all  the  numbers  in  Art.  25. 

UNITED    STATES    MONEY. 

28.  The  money  of  the  United  States  has  a  decimal  system  ; 
dollars  are  written  as  integers,  cents  and  wills  as  decimal 
parts  of  a  dollar.  Instead  of  writing  the  word  dollars  after  a 
number,  we  write  the  '^ dollar  mark''  ($)  before  the  number. 

Thus,  10  dollars  is  written  $10. 

29.  A  point,  called  the  Decimal  Point,  is  placed  at  the 
left  of  the  cents,  but  at  the  right  of  the  dollar  mark. 

Thus,  25  cents  is  written  $.35.     Five  cents  is  written  $.05. 

1.  In  writing  both  dollars  and  cents,  the  decimal  point  is  placed 
between  them. 

Thus,  6  dollars,  15  cents,  is  written  $6. 15. 

2.  The  first  two  places  at  the  right  of  the  decimal  point  are  read  as 
cents,  and  the  third  as  mills. 

Thus,  $4,205  is  read  4  dollars,  20  cents,  5  mills. 


NOTATION   AND  NUMERATION.  23 

30.  Read  the  following  : 

1.  $2.75,       $12.82,       $375.40,       $.87,         |-^30.05,    $100. 

2.  $16.08,     $43.00,       $501.09,       $6.90,       $.54,  $9.86. 

3.  $107.09,  $36.80,       $275,005,     $690.13,  $7.56,      $5,245. 

4.  $260,306,      $150.05,      $309,401,       $3060.40,      $58,039. 

til.  Write  ill  figures  : 

1.  Eight    dolhirs,    ten    cents.      Twelve    dollars.      Eight 
cents.     Nine  dollars,  six  cents. 

2.  Fourteen  cents.     Three   dollars,  sixty  cents.     Fifteen 
dollars,  four  cents.     Five  cents. 

3.  Sixty  dollars,  six  cents.     Two  hundred  ninety  dollars, 
seventy-two  cents.     Six  dollars. 

4.  Five  hundred  one  dollars,  eight  cents.     Nine  hundred 
dollars,  thirty  cents. 

5.  Nine  thousand,  seven  hundred  four  dollars,  forty,  cents. 
Five  dollars,  nine  cents. 

6.  Seven  thousand   four  dollars,   fifty-one   cents.      Sixty- 
five  cents.     Nine  dollars. 

32.  Read  and  name  the  abstract  numbers  : 

1.  $275.  6.7016.  9.  $.05  13.  24  birds. 

2.  $3.24.  6.  $1.  10.  200.  14.  12  feet. 

3.  500.  7.  3  boys.  11.  6  cows.  15.  3250. 

4.  5  cents.  8.  3.  12.  1  man.  16.  3  pounds. 

THE    ROMAN    NOTATION. 

33.  This  system  employs  seven  capital  letters  to  express 
numbers,  viz.: 

I  V  X  L  C  D  M 

1  5  10  50  100         500         1000 

Other  numbers  are  expressed  by  combining  these  letters  in 
accordance  with  the  following 

34.  Prin^ciple. — 1.   Repeating  a  letter  repeats  its  value. 
Thus,  III  represents  three ;  XX,  twenty. 


24 


SCHOOL  ARITHMETIC. 


2..  When  a  letter  precedes  one  of  greater  value,  the  differ- 
ence of  their  values  is  the  number  represented. 

Thus,  IV  represents  the  difference  between  ^ye  and  one,  or  four. 

3.  Wien  a  letter  foUoivs  one  of  greater  value,  the  sum  of 
their  values  is  the  number  represented. 

Thus,  VI  represents  the  sura  of  Jive  and  one,  or  six, 

A  dash  over  a  letter  increases  its  value  a  thousand-fold. 

Thus,  V  represents  five  thousand. 

These  principles  are  illustrated  in  the  following  table  : 


I... 

...1. 

II  .. 

...2. 

III... 

...3. 

IV... 

..A. 

V... 

...5. 

VI... 

...6. 

VII... 

...7. 

VIII... 

...8. 

IX... 

...9. 

X... 

..10. 

XI.. 

XII.. 
XIII.. 
XIV.. 

XV.. 

XVI.. 

XVII. . 

XVIIL. 

XIX.. 

XX.. 


11. 

12. 
13. 
14. 
,15. 
16. 
17. 
18. 
19. 
20. 


XXI 

XXIV 

XXV 

XXIX 

XXX 

XL. 

L, 

LX. 

LXX. 

LXXX. 


.21. 
,24. 
25. 

,29. 
,30. 
,40. 
50. 
60. 
70. 
80. 


xc 

c. 

cc 

ccc 

CD 
D 

DC 
M 

Mx\I 
X 


....90. 
...100. 
...200. 
...300. 
...400. 
...500. 
...600. 
..1000. 
..2000. 
.10000. 


This  method  of  notation  is  called  the  Roman,  because  the  Romans 
invented  and  used  it.  It  was  in  common  use  prior  to  the  introduction  of 
the  Hindu  system,  but  its  use  is  now  very  limited.  Let  the  pupil 
ascertain  by  observation  for  what  purposes  it  is  now  used. 

36.  Express  by  the  Arabic  notation  : 

1.  XXVII,    XVIII,   XLIX,   XLVII,    LVI,    LIX. 

2.  LXII,    LXIX,    LXXV,    LXXXIX,    XCII,    XCVIL 

3.  CVIII,    CXIV,    CCIV,    CCCLXV,    DVII,    DCVII. 

4.  DXLVI,    DXCI,    DCXI,    DCXLIX,    MDCCCC. 

5.  MCDXCII,    MDCCLXXV,  MDCCCXCIIL 
36.  Express  by  the  Roman  notation  : 

1.  18,    27,    36,    45,    54,    63,    72,    81,    94,    97,    101. 

2.  110,  203,  470,  509,  747,  931,  1009,  1865,  1789. 

3.  1607,  1682,  1498,  1620,  1783,  1812,  1901,  2222. 


ADDITION. 

37.  Ill  one  box  I  liave  4  biisliels  of  corn,  in  another  2 
bushels,  in  a  third  3  busliels.  If  I  put  all  into  one  box,  how 
much  corn  will  be  in  that  box  ? 

Here  we  have  three  numbers  to  be  added  (addends),  and 
the  result — 9  bushels — is  the  how  m?(ch,  the  quantity,  which 
we  wished  to  find.     It  is  the  sum  of  the  parts  or  addends. 

38.  The  process  of  finding  the  sum  of  two  or  more  like 
numbers  is  called  Addition. 

1.  What  is  the  sum  of  5  feet,  3  feet,  and  7  feet  ? 

2.  What  is  the  sum  of  6  quarts  and  4  pounds  ? 

3.  Are  6  quarts  and  4  pounds  like  numbers  ? 

4.  Is  the  sum  of  two  or  more  numbers  the  same  in  what- 
ever order  they  are  added  ?    Find  out  by  trial. 

39.  liike  Numbers  are  those  that  have  the  same  unit. 
Thus,  5  yards  and  3  yards    are  like  numbers  ;  but  5  yards  and  3 

pecks  are  not  like  numbers. 

1.  The  sum  of  3  feet  and  G  feet  is  9  what  ? 

2.  Is  the  sum  always  like  the  numbers  added  ? 

40.  Principle. — Onli/  like  mnnhers  can  he  added. 

Find  the  sum  of  : 

1.  3  and  5.  4.  2,  3,  and  6.  7.  G  boys  and  2  boys. 

2.  6  and  2.  5.  3,  5,  and  4.  8.  3  birds  and  G  birds. 

3.  4  and  7.  6.  4,  2,  and  7.  9.  a  cents  and  a  cents. 

41.  The  Sign  of  Addition  (+)  is  called  ^jZ^,"?,  and  is 
placed  between  the  numbers  to  be  added. 

Thus,  5  +  3  is  read  5  plus  3,  and  means  that  3  is  to  be  mlded  to  5. 

42.  The  Sign  of  Equality  ( = )  is  read  equals,  or  is  equal  to. 
Thus,  5  +  3  =  8  is  read  five  plus  three  equals  eight. 


SCHOOL  ARITHMETIC. 


10. 

11. 

12. 
13. 
14. 
15. 


43.  A  statement  that  two  numbers  or  expressions  of  num- 
ber are  equal  is  called  an  Equation. 

Thus,  5  +  3  =  8  is  an  equation, 

44.  Find  the  sum  of  the  following,  and  read  the  complete 
expressions  : 

16.  2  +  8  =  (     ).  31. 

1.7.  2  +  9  =  (     ).  32. 

18.  3  +  3  ==  (     ).  33. 

19.  3  +  4  =  (     ).  34. 

20.  3  +  5  =  (     ).  35. 

21.  3  +  6  =  (     ).  36. 

22.  3  +  7  =  (     ).  37. 

23.  3  +  8  =  (     ).  3&. 

24.  3  +  9  =  (     ).  39. 

25.  4  4-  4  =  (     ).  40. 

26.  4  +  5  =  (     ).  41. 

27.  4  +  G  ==  (     ).  42. 

28.  4  +  7  =  (     ).  43. 

29.  4  +  8  =  (     ).  44. 
2  4-  7  =  (  ).         30.  4  +  9  =  (     ).  45. 

All  the  combinations  of  numbers  from  1  to  9,  taken  two  and  two,  a 
given  in  these  45  equations.  The  pupil  should  practice  adding  the  nui 
bers  until  able  to  state  the  sums  at  a  glance. 

45.  In  blackboard  exercises  various  devices  may  be  used 
to  advantage.  Place  these  diagrams  on  the  blackboard,  and 
indicate  with  a  pointer  the  numbers  to  be  added. 

3  7 

"^       4 


1  +  1  =  (  ). 

l  +  2==(  ). 

l  +  3.=  (  ). 

l  +  4  =  (  ). 

1  +  5  -  (  ). 

l  +  6=(  ). 

7.  1  +  7  =  (  ). 

8.  1   +   8  =:   (  ). 

9.  1  +  9  ==  (  ). 
+  2=(  ). 
+  3-(  ). 
+  4=(). 
+  5  -  (  ). 
+  G  =  {). 


+  5  r= 

+  C  = 

+  7  = 

+  8=. 

5  +  9  = 
G  +  G  = 
G  +  7  = 

6  +  8  = 
G  +  9  = 

7  +  7  = 
7  +  8  = 

7  +  9  = 

8  +  8  = 

8  +  9  = 

9  +  9  = 


5        3        8 

7       4       G 
2       4       9 


1.  Combine  the  numbers  written  at  the  center  and  points 


ADDITION.  27 

of  the   star   into    a    convenient    number   of    exercises;    as 

4  +  7,  5  +  8  4-  6,  etc. 

2.  Add  the  number  in  tlie  center  of  the  square  to  each 
of  the  other  numbers. 

3.  Using  tlie  diagram  at  the  right,  begin  at  a  certain 
number  and  add  in  oitlier  direction  to  50,  100,  etc.  These 
exercises  should  be  continued  until  quick  and  correct  answers 
can  be  given. 

ORAL    EXERCISES. 

46.  1.  Edna  lias  G  bool^s  and  Ida  has  7.  IIow  many  have 
they  both  ? 

2.  Mario  paid  8  cents  for  paper,  5  cents  for  ink,  and  7 
cents  for  pencils.     How  much  did  she  pay  for  all  ? 

3.  James  is  9  years  old,  and  Tom  is  6  years  older.  How 
old  is  Tom  ? 

4.  Six  chickens  are  on  the  fence,  8  are  in  the  barn,  and 

5  are  picking  grass.     How  many  are  there  in  all  ? 

5.  1  paid  $4  for  a  hat,  19  for  a  coat,  and  had  $7  left. 
How  much  had  I  at  first  ? 

6.  A  boy  walked  3  miles  to  the  station,  then  rode  10  miles 
in  the  cars,  and  then  went  8  miles  in  a  coach.  How  far 
did  he  travel  ? 

7.  A  farmer  killed  3  pigs  and  2  sheep,  and  had  9  pigs  and 
4  sheep  left.     How  many  of  both  had  he  at  first  ? 

8.  How  many  days  are  6  days,  5  days,  and  a  week  ? 

9.  How  many  letters  in  the  names  of  all  the  days  of  the 
week  ? 

10.  What  is  the  sum  of  the  numbers  represented  by  the 
ten  digits  ? 

11.  Harry  ran  twice  around  a  house  3  rods  long  and  2 
rods  wide.     How  far  did  he  run  ? 

12.  Mary  has  5  cents,  Kate  has  4  cents,  and  Jane  has  7 
cents  more  than  Kate.  IIow  much  have  Jane  and  Mary  to- 
gether ? 


28  SCHOOL  ARITHMETIC. 

13.  A  lias  $3,  B  has  $5,  and  C  has  as  much  as  both. 
How  much  have  they  all  ? 

14.  One  man  has  a  dollars  and  another  has  2a  dollars. 
How  much  have  both  ? 

15.  Ella  has  2^  cents.  May  has  3^  cents,  and  Tillie  has 
as  many  as  both.     How  many  cents  have  they  all  ? 

16.  What  is  the  sum  of  a,  2«,  ia,  and  5«  ? 

47.  In  the  following  exercise  add  3  to  each  number  in 
column   A,  then  to  each  in  column 
10  4-  3,  20  +  3,  etc.;  then,  31  +  3, 

E 

24 

64 

54 

14 

74 

84 

34 

94     - 

44 

When  the  class  can  announce  these  results  at  sight,  the 
exercise  may  be  varied  by  substituting  one  of  the  other  digits 
for  3.     Using  all  the  digits  gives  729  examples. 

48.    1.  Add  by  2's  from  2  to  50.  From  1  to  49. 

2.  Add  by  3'*s  from  3  to  30.  From  1  to  31. 

3.  Add  by  3's  from  2  to  32.  From  52  to  70. 

4.  Add  by  4's  from  4  to  40.  From  1  to  45. 

5.  Add  by  4's  from  2  to  34.  From  3  to  51. 

6.  Add  by  5's  from  1  to  41.  From  2  to  52. 

7.  Add  by  5's  from  3  to  43.  From  4  to  64. 

8.  Add  by  6's  from  6  to  72.  From  1  to  49. 

9.  Add  by  6's  from  2  to  50.  From  3  to  45. 

10.  Add  by  6's  from  4  to  46.  From  5  to  53. 

11.  Add  by  7's  from  7  to  84.  From  1  to  50. 


A 

B 

C 

D 

10 

31 

52 

83 

20 

41 

92 

53 

60 

81 

12 

73 

80 

91 

42 

23 

30 

21 

62 

13 

90 

51 

32 

63 

50 

71 

82 

43 

40 

11 

72 

33 

70 

61 

*22 

93 

t  B, 

and  so 

on.  '\ 

'hus, 

:1  -f- 

3,  etc. 

F 

G 

H 

I 

95 

46 

67 

78 

35 

16 

77 

88 

45 

36 

27 

98 

75 

66 

57 

38 

55 

86 

97 

48 

15 

76 

47 

28 

25 

96 

17 

68 

65 

26 

87 

58 

85 

56 

37 

18 

ADDITION.  2^ 

12.  Add  by  7's  from  2  to  58.  From  3  to  66. 

13.  Add  by  7's  from  4  to  60.  From  5  to  75. 

14.  Add  by  7's  from  6  to  83.  From  0  to  49. 

15.  Add  by  8's  from  8  to  96.  From  1  to  97. 

16.  Add  by  8's  from  2  to  82.  From  3  to  83. 

17.  Add  by  8's  from  4  to  100.  From  5  to  93. 

18.  Add  by  8's  from  6  to  94.  From  7  to  167. 

19.  Add  by  9's  from  9  to  108.  From  1  to  109. 

20.  Add  by  9's  from  2  to  92.  From  3  to  102. 

21.  Add  9  eight  times  to  4.     To  5.     To  6.     To  7. 

WRITTEN    EXERCISES. 

49.  Copy  and  add  each  column  from  bottom  to  top,  then 

from  top  to  bottom  ;  also  add  each  line,  (as  a),  from  left  to 
right,  then  from  right  to  left. 

In  adding,  do  not  say  7  and  2  are  9,  and  4  are  13,  etc.,  but  give  results 


onl> 

-.    T] 

bus,  7, 

9,13, 

19,  etc. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

a 

5 

3 

6 

i 

5 

9 

8 

7 

4 

9 

3 

b 

3 

6 

5 

4 

7 

6 

4 

6 

8 

4 

8 

c 

6 

4 

rv 

5 

6 

4 

5 

9 

7 

6 

•    7 

d 

4 

7 

3 

9 

4 

7 

9 

0 

6 

5 

6 

e 

2 

8 

4 

3 

9 

8 

7 

8 

3 

7 

9 

f 

7 

5 

9 

8 

8 

5 

3 

4 

9 

8 

5 

12      13      14      15      16      17      18      19      20       21       22 

693889754  7  6 


h 

5 

8 

4 

7 

3 

5 

3 

1 

8 

9 

7 

i 

8 

4 

5 

6 

9 

7 

8 

4 

7 

6 

5 

J 

7 

6 

8 

3 

6 

8 

4 

9 

5 

3 

8 

k 

3 

7 

7 

4 

5 

6 

5 

2 

9 

8 

4 

1 

2 

5 

6 

5 

8 

3 

9 

8 

7 

5 

9 

m 

9 

9 

9 

9 

6 

9 

6 

6 

6 

9 

6 

n 

7 

3. 

7 

5 

7 

5 

7 

7 

8 

2 

8 

0 

4 

5 

8 

7 

5 

0 

6 

8 

3 

7 

7 

30  SCHOOL  ARITHMETIC. 


23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

p 

2 

6 

4 

5 

6 

7 

8 

9 

8 

7 

6 

q 

4 

5 

6 

7 

8 

9 

7 

6 

5 

4 

3 

r 

6 

7 

8 

9 

< 

6 

5 

4 

3 

2 

9 

s 

8 

9 

7 

6 

5 

4 

9 

7 

6 

8 

7 

t 

3 

4 

5 

7 

6 

5 

7 

5 

4 

5 

8 

u 

5 

8 

9 

8 

4 

8 

6 

7 

7 

9 

6 

V 

9 

6 

5 

4 

3 

7 

9 

6 

9 

6 

9 

w 

7 

9 

6 

9 

8 

9 

8 

8 

8 

7 

7 

X 

4 

5 

7 

6 

9 

6 

5 

9 

6 

8 

6 

y 

8 

7 

8 

7 

6 

4 

7 

7 

5 

3 

9 

z 

6 

8 

9 

8 

7 

3 

4 

6 

7 

9 

8 

ORAL   EXERCISES. 

50.  1.  IS'ell  paid  20  cents  for  a  slate  and  30  cents  for  a 
book.     How  much  did  she  pay  for  both  ? 

2.  There  are  30  days  in  April  and  31  in  May.  How  many 
days  in  both  months  ? 

3.  In  an  orchard  there  are  40  apple  trees,  25  peach  trees, 
and  12  pear  trees.  How  many  trees  are  there  in  the  or- 
chard ? 

4.  William  is  12  years  old,  and  his  father  is  27  years 
older.     How  old  is  his  father  ? 

5.  How  many  days  are  there  in  May,  June,  and  July  ? 

6.  If  Edgar  has  $25,  and  Allen  has  120,  and  I  have  $15 
more  than  they  both  have,  how  much  have  I  ? 

7.  An  army  marched  25  miles  one  day  and  4  miles  far- 
ther the  next  day.     How  far  did  it  march  in  both  days  ? 

8.  A  drover  sold  25  sheep  to  one  man  and  42  to  another, 
and  had  23  left.     How  many  had  he  at  first  ? 

9.  A  field  is  40  rods  long  and  16  rods  wide.  What  is 
the  distance  around  the  field  ? 

10.  A  newsboy  sold  40  morning  papers,  25  evening  papers, 
and  had  12  left  on  his  hands.     How  many  did  he  buy  ? 


ADDITION.  31 

11.  A  dealer  sold  25  bushels  of  coal  to  one  man,  35  to 
another,  and  32  to  another.  How  many  bushels  did  he  sell 
to  all  ? 

12.  After  paying  $33  for  a  cow  and  $51  for  a  wagon,  I  had 
left  a  ten-dollar  bill,  a  five-dollar  bill,  and  a  silver  dollar. 
How  much  money  had  I  at  first  ? 

13.  Forty-one  years  ago  Jane's  grandpa  was  36  years  of 
age.     How  old  will  he  be  if  he  lives  5  years  longer? 

14.  A  lady  bought  3  webs  of  cloth,  the  first  containing 
32  yards,  the  second  41  yards,  and  the  third  36  yards.  How 
many  3^ards  did  she  buy  ? 

15.  On  Monday  Ada's  teacher  gave  her  20  words  to 
spell,  on  Tuesday  25,  on  Wednesday  25,  on  Thursday  30, 
and  on  Friday  18.  How  many  words  were  given  in  the 
week  ? 

16.  Willie  has  a  marbles,  Harry  has  4«,  and  Rob  has  as 
many  as  both.     How  many  have  they  all  ? 

17.  A  man  had  a  pigs  and  bought  b  more.  How  many 
did  he  then  have  ?     («  +  b). 

18.  Sarah  has  a  cents,  Alice  has  b  cents,  and  Ellen  has 
as  many  as  Sarah.     How  many  have  they  all  ? 

19.  What  is  the  sum  of  «,  da,  da,  and  2b  ? 

20.  If  one  stone  weighs  ])  pounds  and  another  weighs  q 
pounds,  what  is  the  weight  of  both  together  ? 

WRITTEN     EXERCISES. 

51.  1.  What  is  the  sum  of  7597,  1368,  and  643  ? 
7597  May  the  numbers  be  added  without  placing  them  in  vertical 

1368     columns?    Which  way  is  the  more  convenient? 

643  Are  the  tens  all  in  the  same  column?    In  which  column  are 

ggQg     the  ones?    The  hundreds? 

What  is  the  sura  of  the  ones?    Where  is  the  8  written?  What 
is  done  with  the  1  ten? 

What  is  the  sum  of  the  tens?  What  is  done  with  theO?  With  the 
2  hundreds?    Explain  bow  the  9  thousands  is  obtained. 


372  =  300  + 
458  =  400  + 

765  =  700  -f 

70  + 
50  + 
60  + 

2 

8 
5 

1595  =  1400  + 
or  1500  + 

180  + 
90  + 

15 
5 

32  •  SCHOOL  ARITHMETIC. 

2.  Find  the  sum  of  372,  458,  and  765. 

We  here  see  that  the  sum  of  the  ones 
is  15  ;  the  sum  of  the  tens  18,  or  180 
ones  ;  the  sum  of  the  hundreds  14,  or 
1400  ones.  Since  15  =  1  ten  and  5  ones, 
we  have  18  +  1,  or  19  tens,  or  190  ones. 
And  since  190  =  1  hundred  and  90  ones 
(9  tens),  we  have  15  hundreds,  or  1500  ones.  Hence,  1500  +  90  +  5  = 
1595,  the  sum. 

Direction. — Arrange  the  numbers  so  that  units  of  the  same 
order  shall  stand  in  the  saine  colufnn. 

Begin  at  the  right  and  add  each  column  separately.  If  the 
sum  is  less  than  10,  write  it  tinder  the  column  added.  If  it 
is  10  or  more,  place  only  the  right-hand  figure  under  that 
column,  and  add  the  remainder  with  the  next  column. 

To  insure  accuracy,  add  each  column  from  top  downwards 
as  well  as  from  hottoin  upwards. 

Write  from  dictation  and  add  : 
3  4  5  6  7  8 

476  418  374  293  416  567 

358  736  627  431  907  891 

924  375  258  827  692  234 

265  632  196  546  375  432 

When  long  columns  are  to  be  added,  the  following  method 
will  be  found  practical.     Explain  the  process. 

327 

841 

576 

439 

765 

192 

30 

31 

28 

3140 


ADDI 

riON. 

33 

Add 

in  two  ininu 

tes  : 

9 

10 

11 

12 

13 

14 

oG43 

1234 

9876 

9 

568 

6677 

7328 

5678 

5432 

87 

7329 

896 

3576 

9123 

2347 

654 

8695 

54 

8794 

4567 

678 

3210 

87 

7 

4887 

8923 

4706 

123 

9764 

88 

6452 

4567 

934 

45 

9 

945 

7968 

9876 

87 

6 

846 

8776 

5679 

5432 

5878 

5000 

6375 

9438 

9785 

7777 

6660 

9999 

8888 

7897 

Find  the  value  of  the  following  : 

15.  728  +  2076  +  7583  +  7127  +  928  +  8267. 

16.  83267  +  48  +  2498  +  6735  +  9476  +  10783. 

17.  9847  +  3084  +  86540  +  4306  +  89  +  9008. 

18.  63075  +  350  +  4703  +  3715  +  937  +  15008. 

19.  16160  +  6048  +  260  +  4826  +  50598  +  8763. 

20.  357  +  913  +  579  +  135  +  791  +  638  +  794  +  867. 

21.  5894  +  876  +  93  +  4672  +  8975  4-  8456  +  9784. 

22.  9768  +  87  +  69  +  764  +  893  4-  5768  +  88  +  8769. 

23.  Add  8568,  3864,  6846,  5976,  7249,  4839. 

24.  Add  3925,  6868,  4857,  8394,  8426,  9397. 

25.  Add  3987,  4876,  9254,  7983,  8427,  2935. 

26.  Add  5763,  3854,  87,  609,  975,  4508,  7358. 

27.  Add  4978,  9834,  734,  5627,  8,  3764,  47,  835. 

28.  Add  6984,  8592,  5807,  74,  96,  8958,  64,  789. 

29.  Add  5926,  7859,  4768,  8729,  384,  769,  8943. 

30.  Add  8592,  5678,  6854,  4673,  7968,  6843,  4396,  78. 

31.  Add  476,  5814,  6820,  21,  657. 

32.  Add  5215,  4863,  9211,  20781. 

33.  Add  48'i2,  9712,  48,  63,  14. 

34.  Add  97,  86741,  2248,  6127,  487,  57120. 

35.  Add  282,  4789,  63175. 

3 


34 


SCHOOL  ARITHMETIC. 


RAPID    ADDITION. 


In  rapid  addition  various  methods  are  employed  by  ac- 
countants, one  of  the  most  commonly  used  being  what  may 
be  called  the  '^grouping  method/'  which  is  here  briefly 
illustrated. 


*(a) 

(b)           (c) 

W 

(e) 

(f) 

(g) 

(h) 

5-| 

91         r  6'] 

9" 

J8 

7 

4 

7 

2 

iJ            iJ 

3_ 

(9" 

3 

6 

5 

3- 

8           (7 

8~ 

(^-- 

1 

9 

2 

7" 
3 

8 

6-1        (4" 

1            ]o 
3J        (3- 

6_ 
9_ 

(5J 

6 
4 
2 

2 
3 

7 

8 
6 
9 

6- 
4_ 

^1         j^- 
Sj         18 

5" 

7_ 

is-] 

6_ 

4 
2 

1 
4 

4 
6 

38 

"37       Yo 

51 

51 

8 

5 

8 

In  adding  column  (a),  we  may  group  two  or  more  numbers 
whose  sum  is  10,  and  add  thus  :  10,  18,  28,  38. 

In  (b)  the  sum  may  be  10  or  less,  and  we  say,  9,  19,  27,  37. 

In  (c)  we  may  say,  8,  17,  26,  33,  40.  Or,  making  the  sum 
greater  than  10,  we  may  add  thus :  14,  26,  40. 

In  (d)  we  may  proceed  thus  :  12,  25,  39,  51. 

In  (e)  our  count  may  be,  14,  27,  43,  51  ;  or,  beginning  at 
the  top,  17,  32,  45,  51. 

*  For  the  first  practice  in  grouping,  the  teacher  should  provide  num- 
bers that  readily  fall  into  groups  whose  sum  is  10,  as  in  (a).  These 
exercises  should  be  followed  by  others  where  the  Sum  is  10  or  less,  as  in 
(b),  and  these  in  turn  by  more  difficult  ones  where  the  sum  of  the  group 
is  from  10  to  19. 

As  a  preparation  for  this  work,  the  pupil  should  be  given  abundant 
practice  in  announcing  at  sight  the  sum  of  any  two  numbers  between  10 
and  20  ;  as  11  +  11,  11  +  12,  12  +  13,  etc.  There  are  81  of  these  com- 
binations. 

Exercises  in  rapid  addition  should  be  regular,  not  spasmodic.  Sur- 
prising results  may  be  obtained  in  a  single  term  by  devoting  ^I'e  minutes 
each  day  to  intelligent  practice. 


ADDITION.  35 

36.  Try  to  group  the  mimbers  in  (f).    Which  do  you  find  the 
better  place  to  begin — top  or  bottom  ? 

37.  Group  the  numbers  in  (g),  and  add. 

38.  Group  the  numbers  in  (h)  in  as  many  ways  as  you  can. 
AVhicli  way  is  best  ? 

DOUBLE-COLUMN   ADDING. 

Many  persons   add    two   columns    at    once,    some    three 
columns.     The  process  is  as  follows  : 

5731  Beginning  at  the  bottom  we  add  tens  and  ones,  saying  (or 

3287       thinking),  6-4,  10-7,  16-3  (that  is,  15-13),  19-2  (or  18-1*2),  27-9, 
31-0  (or  30-10).     The  sura  is  31  tens  and  no  ones. 

When  we  get  10  or  more  ones,  we  "carry"  the  1  ten  over 

to  the  tens  cohimn.      Thus,  when  our  sums  are  15  tens-13 

ones,  we  say  16-3.     In  this  way  we  avoid  having  to  remember 

7664       more  than  one  figure  in  tlie  right-hand  cohimn. 

^^77^  Taking  tlie  second  two  columns  we  count  thus  :   7-6,  14-5, 

^^^       19-3,   27-9,   31-1,    36-8.      The    sum    is   36  thousands  and  8 

3^8  hundreds. 

37110  ^^  before,  whenever  we  have  a  sum  of  10  or  more  in  the 

right-hand  column,  the  1  ten  is  added  with  the  other  column. 
Thus,  instead  of  13-15  we  have  14-5  ;  instead  of  18-13  we  have  19-3. 

Add  each  of  the  following  in  two  minutes  : 

39.  40.  41. 

12345  (>  78  1234567890  78517  6  89 

7486  3  925  987654  5  6  78  56395467 

57687436  2357898765  3  417  3245 

83796894  4  2  13456843  129  5  102  3 

46878513  87  6  5987  6  54  89628790 

89454789  3989739437  563954  6  7 

65987665  876  6  874989  2  3  062134 

5  4668937  7898765346  9573987  8 

787  9  9876  6  3  74942105  7788699  2 

3  784  5  498  8387659872  86947687 

9957  6  747  9876543218  68598769 

678987  6  5  87  6  5466789  99679578 


8629 
4856 
6943 


8g  SCHOOL  ARITHMETIC. 

41  Add  382  thousand,  4  hundred  53  ;  514  thousand,  6 
hundred  85  ;  684  thousand,  3  hundred  25 ;  298  thousand, 
5  hundred  76  ;  176  thousand,  7  hundred  92. 

43.  Add  24  million,  356  thousand,  8  hundred  13  ;  92 
million,  75  thousand,  3  hundred  46  ;  7  million,  310  thousand, 
1  hundred  6  ;  30  million,  30  thousand,  3  hundred  30  ;  8 
million,  8  thousand,  8  hundred  8. 

44.  Add  376  million,  724  thousand,  9  hundred  86;  4 
million,  315  thousand,  8  ;  591  million,  304  thousand,  81  ;  79 
million,  58  thousand,  627  ;  83  million,  726  ;  819  million,  10 
thousand,  50. 

45.  Add  six  million,  two  thousand,  five  hundred  forty- 
one  ;  eight  million,  seven  hundred  thirty-eight  thousand,  four 
hundred  three;  one  million,  seven  thousand,  nine;  thirty 
million,  eighty-nine  thousand,  fourteen;  five  hundred  nine 
thousand,  eight  hundred  thirty. 

46.  Find  the  sum  of  two  million,  nine  thousand,  foilr 
hundred  seventy-six  ;  seven  million,  forty  thousand,  sixteen  ; 
three  million,  twenty-four ;  nine  hundred  three  thousand, 
ten  ;  six  million,  six  thousand,  six  hundred  six. 

47.  A  railroad  train  ran  376  miles  on  Monday,  298  miles 
on  Tuesday,  437  miles  on  Wednesday,  326  miles  on  Thursday, 
265  miles  on  Friday,  368  miles  on  Saturday,  and  20  miles  on 
Sunday.     How  many  miles  did  it  run  in  the  week  ? 

48.  In  one  year  a  man  pays  $375  for  rent,  $537  more  than 
that  for  other  expenses,  and  has  $513  left  out  of  his  salary. 
What  is  his  salary  ? 

49.  A  farmer  sold  289  bushels  of  corn  to  one  man,  397  to 
another,  568  to  another,  197  to  another,  and  then  had  685 
bushels  left.     How  many  bushels  had  he  at  first  ? 

60.  A  man  owns  five  lots.  The  first  cost  $325,  the 
second  $275,  the  third  $450,  the  fourth  $580,  and  the  fifth 
$240  more  than  the  other  four.  How  much  did  they  all 
cost? 

51.  The  area  of   New  York  is  49,170  square   miles;    of 


ADDITION.  37 

Pennsylvanhi,  45,215;  of  New  Jersey,  7,815;  of  Delaware, 
2,050.     What  is  the  entire  area  of  these  four  States  ? 

62.  A  mail  traveled  328  miles  a  day  for  three  days,  and 
276  miles  a  day  for  the  next  three  days.  How  far  did  he 
travel  in  the  six  days  ? 

53.  The  first  year  a  man  worked  in  an  iron  mill  he  re- 
ceived $450.  If  his  salary  was  increased  $125  a  year,  for 
four  years,  how  much  did  he  earn  in  the  five  years  ? 


54.  In  a  stock-yard  there  are  as  many  cows  as  calves,  and 
as  many  sheep  as  hogs.  If  there  are  G8  calves  and  97  hogs, 
how  many  animals  are  in  the  yard  ? 

55.  The  school-yard  fence  has  289  pickets  on  the  front, 
and  an  equal  number  on  the  back.  On  each  end  there  are 
257.     How  many  pickets  are  there  on  the  whole  fence  ? 

56.  One  side  of  a  square  farm  is  2854  feet  long.  What 
is  the  distance  around  the  farm  ? 

57.  A  book-case  has  six  shelves.  The  first  two  contain 
28  books  each,  the  second  two  34  each,  and  the  last  two  39 
each.     How  many  books  are  there  on  the  six  shelves  ? 

58.  A  merchant  sold  goods  to  the  amount  of  $485  on 
Monday,  and  during  the  remainder  of  the  week  he  increased 
his  sales  $98  each  day.     What    was  the  total  value  of  his 

*  week's  sales  ? 

59.  A  girl  has  a  dollars,  her  brother  has  h  dollars  more 
than  she  has,  and  their  father  has  h  dollars  more  than  both. 
How  much  have  all  three  ? 

^  60.  A  train  makes  the  round  trip  between  Philadelphia  and 
Pittsburg  every  two  days.  If  the  distance  between  these 
cities  is  354  miles,  how  many  miles  does  the  train  run  in  4 
days  ? 

61.  Mr.  A  bought  a  lot  for  S875,  and  sold  it  so  as  to  gain 
$750.     At  what  price  did  he  sell  ? 

62.  A  man  owning  a  large  tract  of  land  divided  all  of  it 


38  SCHOOL  ARITHMETIC. 

except  160  acres  amoDg  his  4  sons  and  3  daughters.  To  the 
eldest  son  he  gave  320  acres,  and  to  each  of  the  others  he 
gave  180  acres.  To  each  of  his  daughters  he  gave  240  acres. 
How  many  acres  were  in  the  tract  ? 

63.  The  fiumber  of  lobsters  caught  in  1889  was  as  follows; 
Maine,  12,552,866;  Massachusetts,  2,624,218;  Connecticut, 
687,994;  Rhode  Island,  538,315;  New  Hampshire,  176,733; 
New  York,  206,875  ;  New  Jersey,  74,866  ;  Delaware,  3,750. 
How  many  were  caught  ? 

64.  The  peanut  crop  of  Virginia  in  1890  was  1,171,624 
bushels ;  of  West  Virginia,  39  bushels  ;  of  Georgia,  624,528 
bushels  ;  of  Pennsylvania,  22  bushels  ;  of  New  York,  106 
bushels  ;  of  Illinois,  481  bushels  ;  of  Tennessee,  523,088 
bushels  ;  of  North  Carolina,  421,138  bushels.  What  was 
their  combined  production  ? 

65.  In  the  year  ending  January  1,  1895,  Kentucky  pro- 
duced 183,618,425  pounds  of  tobacco  ;  Virginia,  35,593,984 
pounds  ;  Ohio,  32,468,938  pounds  ;  Massachusetts,  3,449,655 
pounds  ;  Maryland,  7,010,380  pounds  ;  and  West  Virginia, 
2,634,585  pounds.  How  many  pounds  did  all  six  States 
produce  ? 

66.  The  bank  clearings  June  11,  1900,  in  Charleston,  S.  C, 
were  as  follows  :  People's  National  Bank,  132,125  ;  Bank  of 
Charleston,  $30,219  ;  South  Carolina  Loan  and  Trust  Co., 
$21,041 ;  First  National  Bank,  $25,968.  What  was  the  total 
clearinofs  of  these  banks  ? 


SUBTRACTION. 

62.  1.  If  Kate  has  $5  and  spends  $3,  how  much  has  she 
left? 

2.  How  many  apples  are  left  when  4  apples  are  taken  from 
6  apples  ? 

3.  Seven  marbles  are  how  many  more  than  5  marbles  ? 

4.  What  number  of  cents  added  to  5  cents  will  make  9 
cents  ? 

5.  How  many  dollars  are  left  when  $5  are  taken  from  $8  ? 

53.  The  number  that  is  left  when  one  number  is  taken 
from  another  is  called  the  Remainder  or  Difference. 

54.  The  process  of  finding  the  remainder  or  the  difference 
between  two  like  numbers  is  called  Subtraction. 

1.  Can  two  yards  be  taken  from  3  pounds  ?     Why  not  ? 

2.  What  is  the  remainder  when  $2  is  taken  from  $5  ? 
How  does  $5  compare  with  the  sum  of  $2  and  this  re- 
mainder ? 

3.  Subtract  one  number  from  another  and  see  what  the 
remainder  added  to  the  smaller  number  equals. 

55.  Pkinciples. — 1.  JVumbers  can  be  subtracted  from  like 
numbers  only. 

2:  The  larger  number  is  equal  to  the  sum  of  the  remainder 
and  the  smaller  number. 

56.  The  larger  number,  or  the  one  from  which  another  is 
subtracted,  is  called  the  Minuend. 

57.  The  smaller  number,  or  the  one  subtracted^  is  C9;lled 
the  Subtrahend, 


40 


SCHOOL  ARITHMETIC. 


58.  The  Sign  of  Subtraction  (  — )  is  called  7ninus,  and 
is  placed  before  the  number  to  besnbtracted. 

Thus,  5  —  3  is  read  5  minus  3,  and  means  that  3  is  to  be  subtracted 
from  5. 


59.  Practice    should   be 
exercises  until  pupils  can  gi 


1.  1  +  (  ; 

2. 

2.  2  -     1 

=^ 

(     ). 

3.  1  -f  (   ; 

)  = 

3. 

4.  3  -     2 

=z 

(     >• 

5.  3  -     1 

= 

(     )• 

6.  1  +  (     ) 

:= 

4. 

7.  4-     3 

=1 

(     ). 

8.  4  -     2 

= 

(     ). 

9.  4  -     1 

z=z 

(     ). 

10.  1  +  (    ; 

= 

5. 

11.  5-4 

= 

(     ). 

12.  2  +  (     , 

=. 

5. 

13.  5  -     3 

= 

(  )■ 

14.  5  -     2 

— 

(  )• 

15.  6  -    5 

■=. 

(  )• 

16.  G  -  (     ) 

= 

2. 

17.  6  -    3 

= 

(     )• 

18.  (3  -  (     ) 

=: 

5. 

19.  7  -    G 

- 

(     )• 

20.  7  -  (     ) 

— 

2. 

21.  7  -  (     ) 

■= 

3. 

22.  7  -    3 

=: 

(     )• 

23.  7  -    2 

= 

(     )■ 

24  8  -    7 

z=z 

(     )• 

25.  8  -  (     ) 

= 

2. 

26.  8  -  (     ) 

zz: 

S. 

27.  8  -    4 

= 

{     ). 

28.  {   )-  3 

= 

5. 

continued   upon    the   following 
ve  quick  and  correct  answers. 

29.  8  -  (     )  =:    6. 

30.  9  -    8     =(     ). 

31.  9  -  (     )  =    2. 

32.  9  -    6     =  {     ). 

33.  9  -  (     )  =    4. 
34  (    )-  4     =.      5. 

35.  9  -    3     =  i     ). 

36.  9  -  (     )  =    7. 

•    37.  10  -     9    =  (     ). 

38.  10  -  (     )  =    2. 

39.  10  -     7     = 

40.  10-6     =r 

41.  10  -  (     )  = 

42.  10  -     4     = 

43.  10  -       3       rr 

44    11   -       9       nr 

45.  11  -     8     = 

46.  11  -  (     )  = 

47.  11  -  (     )  :^ 

48.  12  -     9     = 

49.  12  -     8     = 

50.  12  -     7     = 
61.  12  -  (     )  = 

52.  13  -    9     r= 

53.  13  -    8     = 
54  13  -    7     == 

55.  13  -  (     )  =r    7. 

56.  13  -    5     ={     ). 


6. 


SUBTRACTION,  41 

67.  13  -  (     )  ==    9.  65.  15  -  (     )  =    8. 

58.  14  -    9     =  (     ).  66.  15  -    G     =  (     ). 

59.  14  -     8     =  (     ).  67.  10  -    9     =(     ). 

60.  14  -  (     )  =    7.  eS.  16  -  {     )=z    8.       . 

61.  14  -    G     =(     ).  69.  IG  -    7     =(     ). 

62.  14  -  (     )  ==    9.  70.  17  -    9     =  (     ). 

63.  15  -    9     ={     ).  71.  17  -  (     )  =    9. 

64.  15  -    8     ={     ).  72.  18  -    9     =  (     ). 

73.  Subtract  by  2's  from  30.  From  31  to  1. 

74.  Subtract  by  3's  from  3B.  From  34  to  1. 

75.  Subtract  by  4's  from  40.  From  35  to  1. 

60.  In  tl)e  following  exercises  name  only  sums  and  differ- 
ences.    Thus,  in  the  first  example,  say,  8,  12,  7,  13,  6,  1. 

Find  the  value  of  : 

1.  8  +  4-5  +  6-7-5.  7.  8  +  9-4-7  +  12-9 

2.  7-3  +  8-5-4  +  G.  8.  17-8  +  G-7  +  9-8 

3.  4  +  5-G  +  8-7  +  9.  9.  7  +  9-8  +  5-7  +  14 

4.  9-7  +  G  +  5-8-3.  10.  G  +  8-9  +  6-4  +  17 

5.  3  +  8-6  +  9-G-5.  •       11.  5  +  11-7-5  +  9-8 

6.  7  +  6  +  2-6  +  7-9.  12.  9-7  +  8  +  6-7  +  11 

61.  In  the  following  mime  only  remainders,  thus  :  46,  43 
38,  36,  etc. 

1.  50 -4 -3 -5-2-4-3-5-6-4- 3 -2-5  =  (  ) 

2.  50 -5-4-2- 6 -4 -7-5-3-5-2- 3 -4:=(  ) 

3.  50  -  3  -  5  -  5  -  3  -  2  -  3  -  5  -  5  -  6  -  4  -  7  -  2  r=  (  ) 

4.  50 -6-3-4-5-4 -6-7-4- 3-5 -2-l  =  (  ) 

5.  50 -7-4-5-6- 3 -4-2-3-5-4-3 -3=.(  ) 

6.  50-3-2-8-5-4-6-l-9-3-4-3-2i:=(  ) 

62.  When  several  numbers  are  included  in  a  Parenthesis 

(     ),  they  are  to  be  treated  as  a  single  number. 

Thus,  (5  +  3)  —  (8  —  G)  means  that  the  difference  between  8  and  6  is 
to  be  taken  from  the  sum  of  5  and  3. 

The  parenthesis  as  a  sign  of  aggregation  was  first  used  by  Girard  in 
1629. 


7. 

4  +  5 

+  6- 

(8  -  3). 

8. 

3  +  7  +  (6- 

■  4)  -  5. 

9. 

5  +  8 

-(9  + 

■7-6). 

10. 

(16- 

9)  +8 

-  (12  - 

5). 

11. 

(16- 

9)  +  8 

+  (12- 

5). 

42  SCHOOL  ARITHMETIC. 

63.  Find  the  value  of  : 

1.  8  +  5  -  (3  +  6). 

2.  7  +  9  +  (G  -  4). 

3.  15  -  (2  +  7)  -  4. 

4.  (17-9)-(13-7).       . 

5.  (16  -  7)  +  6  -  8. 

6.  15-6  +  5-7.  12.  (18  -  4)  -  (13  -9  +  8). 

13.  Willie  bought  12  marbles  and  gave  5  of  them  to  his 
brother.     How  many  did  he  keep  ? 

14.  Mary  earns  112  a  month  and  spends  17.  How  much 
does  she  save  a  month  ? 

15.  Tillie  has  9  cents  and  Lottie  has  4.  How  many  more 
cents  has  Tillie  than  Lottie  ? 

16.  Howard  wants  to  buy  a  book  for  17  cents,  but  has 
only  9  cents.  How  many  cents  must  he  get  before  he  can 
buy  the  book  ? 

17.  Tv/o  pieces  of  cloth  contain  16  yards.  If  there  are  7 
yards  in  one  piece,  how  many  arg  in  the  other  ? 

18.  In  a  class  of  14  boys  and  girls  there  are  5  boys.  How 
many  girls  are  in  the  class  ? 

19.  Henry  paid  $19  for  a  cart  and  sold  it  for  $9.  How 
much  did  he  lose  ? 

20.  If  a  farmer  avIio  had  19  sheep  sold  11,  how  many  had 
he  left  ? 

21.  Mrs.  E  had  two  ten-dollar  bills,  and  spent  $4  for  shoes 
and  $8  for  a  dress.     How  much  money  had  she  left  ? 

22.  A  farmer  sold  7  pigs  and  2  more  died.  If  he  had  11 
left,  how  many  had  he  at  first  ? 

23.  Carl  is  11  years  old  and  his  sister  is  7  years  younger. 
How  old  is  his  sister  ? 

24.  Sarah  is  6  years  old  and  her  brother  is  14.  What  is 
the  difference  in  their  ages  ? 

25.  Alice  has  12  plants,  of  which  7  are  geraniums  and  the 
remainder  pinks.     How  many  pinks  has  she  ? 


SUBTRACTION.  43 

26.  A  hunter  shot  15  rabbits  and  squirrels.  If  he  shot  5 
rabbits,  how  many  squirrels  did  he  shoot  ? 

27.  A  man  has  $19  in  gold,  silver,  and  paper.  If  he  has  4 
silver  dollars  and  5  paper  dollars,  how  much  has  he  in  gold  ? 

28.  A  boy  has  18  rows  of  potatoes  to  hoe.  In  the  fore- 
noon he  hoed  0  rows,  in  the  afternoon  7  rows,  and  finished 
the  next  day.     How  many  rows  did  he  hoe  the  next  day  ? 

29.  A  has  19  acres  of  land  and  B  has  9  acres.  How  many 
acres  must  B  purchase  from  A  so  that  B  may  have  10  acres 
more  than  A  ? 

30.  Harry  had  17,  Carrie  had  *9,  and  Ella  had  $8  less 
than  both  together.     How  much  had  they  all  ? 

31.  A  lady  who  had  18  chickens  sold  5  to  one  man  and 
8  to  another,  and  gave  a  pair  to  her  sister.  How  many  had 
she  left  ? 

32.  A  farmer  having  7  horses  sold  them,  and  bought  9  from 
one  man  and  G  from  another.  .  He  afterwards  sold  8,  and  1 
died.     How  many  had  he  then  ? 

33.  James  had  oa  cents  and  spent  a  cents.  How  many 
cents  had  he  left  ? 

34.  AYarren  having  bb  marbles  lost  2b  of  them.  How 
many  had  he  left  ? 

35.  Hattie  is  a  years  old  and  Martha  is  b  years  old.  What 
is  the  difference  in  their  ages  ?     (a  —  b). 

36.  Walter  has  a  dollars,  George  has  2a  dollars,  and  Albert 
has  b  dollars  less  than  both.     How  many  dollars  has  Albert  ? 

37.  A  lady  who  had  7a  hens  sold  a  hens  to  one  man  and  4a 
hens  to  another.     How  many  had  she  left  ? 

38.  Nettie  weighs  a  pounds,  Ida  weighs  b  pounds  less,  and 
Mabel  weighs  a  pounds  more  than  both.  What  is  MabeFs 
weight  ? 

39.  Find  the  value  of  a  +  3a  -\-  a  —  4:a  +  b. 

40.  AVhat  is  the  value  of  6a  +  (3«  —  a)  —  4:a  —  2b  ? 

41.  A  man  having  16  sheep  sold  9,  and  after  buying  some 
more  had  13.     How  many  did  he  buy  ? 


44  SCHOOL  ARITHMETIC. 

42.  Tom  was  5  years  old  10  years  ago.     How  old  was  he  5 
years  ago  ? 

43.  Jerry  was  a  years  old  J  years  ago.     How  old  was  he  a 
years  ago  ? 

44.  David  will  be  18  years  old  7  years  hence.     How  old  was 
he  7  years  ago  ? 

WRITTEN     EXERCISES. 

64.  1.  What  is  the  difference  between  978  and  435  ? 
978  Is  it  necessary  to  write  the  smaller  number  under  the  larger  ? 

435     Is  it  convenient  ? 

w  .„  Under  what  is  the  5  ones  written?    Where  is  the  3  tens  placed? 

The  4  hundreds? 

5  ones  from  8  ones  leaves ones;  3  tens  from  7  tens  leaves 

tens;  4  hundreds  from  9  hundreds  leaves hundreds.     The  required 

difference  is  543, 


Query. — Is  the  sum  of  543  and  435 

equal  to  978?  Does  that  prove 

the  work  to  be  correct  ? 

Find  the  differences,  and  prove 

: 

2.         3. 

4.         6. 

84327       76594 

69748       $86.75 

53124       43251 

27413        14.32 

6.          7. 

8. 

8374967H      9997785( 

)      66887795 

51427343      05424351 

L      34465672 

9. 

10. 

9668987957845 

8379568978694 

4357534825532 

1234567876543 

11. 

8978564837695874968977896708 

3526153726460530733845421602 

Find  the  valne  of  : 

12.  9889  -  6345  +  234. 

17.  7968  -  6725  +  3009. 

13.  7658  -  324  -  5123. 

18.  2436  +  7432  -  6143. 

14.  9768  -   43  -  6402. 

19.  3254  +  435  -  634. 

15.  6995  -  4070  -  825. 

20.  8776  -  4345  -  2231. 

16,  5683  -  5461  +  1012. 

21,  9000  +  678  -  7075. 

SUBTRACTION.  45 

22.  Mr.  C  bought  a  lot  for  $5375  and  sold  it  for  $6798. 
How  much  did  he  gain  ? 

23.  A  farmer  who  raised  876  bushels  of  corn  kept  243 
bushels  for  his  own  use  and  sold  the  remainder.  How  many 
bushels  did  he  sell  ? 

24.  A  man  was  born  in  1873  and  married  in  1898.  At 
what  age  did  he  marry  ? 

25.  A  contractor  agreed  to  build  a  house  for  $5875.  If  his 
expenses  were  $3725,  how  much  did  he  make  ? 

26.  In  a  school  of  2879  pupils  there  are  1324  boys.  How 
many  girls  are  in  the  school  ? 

27.  A  man  gave  a  wagon  and  $135  for  a  horse  worth  $189. 
How  much  was  he  allowed  for  his  wagon  ? 

28.  A  and  B  were  198  miles  apart.  They  traveled  toward 
each  other  one  day,  A  going  42  miles  and  B  35  miles.  How 
far  apart  were  they  then  ? 

29.  Twelve  years  ago  Ella^s  grandpa  was  59  years  of  age. 
How  old  will  he  be  if  he  lives  till  1909  ? 

30.  Tom  ran  around  a  barn  45  feet  long  and  23  feet  wide, 
and  Pat  ran  287  feet.  How  much  farther  did  Pat  run  than 
Tom  ran  ? 

ORAL    EXERCISES. 

65.  1.  Subtract  by  5's  from  49  to  4.     From  47  to  2. 

2.  Subtract  by  6's  from  42  to  6.     From  41  to  5. 

3.  Take  7  from  58  eight  times.     From  60. 

4.  Take  8  from  49  six  times.     From  51. 

5.  Subtract  by  9's  from  54  to  0.     From  64  to  1. 

6.  Count  by  7's  from  2  to  65,  and  back  from  65  to  2. 

7.  How  many  are  4  tens  less  3  tens  ?    40  —  30  ? 

8.  How  many  are  7  tens  less  4  tens  ?     70  —  40  ? 

9.  How  many  are  8  tens  less  3  tens  ?     80  —  30  ? 

10.  How  many  are  9  tens  less  6  tens  ?     90  —  60  ? 

11.  If  you  buy  a  dozen  eggs  for  17  cents,  how  much  change 
should  you  get  out  of  a  quarter  ? 


46  SCHOOL  ARITHMETIC. 

12.  I  paid  $27  for  a  chair  and  a  clock.  If  the  cliair  cost 
$18,  what  did  the  clock  cost  ? 

13.  From  a  cask  containing  25  gallons,  IG  gallons  were 
drawn.     How  many  gallons  remained  ? 

14.  An  agent  purchased  books  for  $17  and  sold  them  for 
$32.     How  much  did  he  gain  ? 

15.  A  farmer  sold  a  cow  for  $41,  which  was  $13  more  than 
the  cost.     Find  the  cost.  ;  ,. 

16.  James  earns  $35  a  month  and  spends '$17.  How  much 
does  he  save  monthly  ?  ..^ 

17.  A  man  having  49  sheep  sold  19  and  killed  11.  How 
many  had  he  left  ? 

18.  A  farmer  having  21a  sheep  sold  13a  and  killed  2a. 
How  many  had  he  left  ? 

19.  When  the  minuend  and  the  remainder  are  given,  how 
can  the  subtrahend  be  found  ?     Why  ? 

20.  If  the  minuend  is  7a,  and  the  remainder  2a,  what  is 
the  subtrahend  ? 

21.  When  the  remainder  and  subtrahend  are  given,  how  is 
the  minuend  found  ?     Why  ? 

22.  If  the  remainder  is  h  and  the  subtrahend  5b,  what  is 
the  minuend  ? 

23.  Mr.  B  had  a  twenty-dollar  bill,  a  ten-dollar  bill,  and 
a  five-dollar  bill.  He  bought  a  hat  for  $6  and  a  coat  for 
$13.     How  much  money  had  he  left  ? 

WRITTEN     EXERCISES. 

66.  1.  Subtract  375  from  632. 

632  Since  we  cannot  take  5  ones  from  2  ones,  we  add  1  of  the  3 
375  tens,  or  10  ones,  to  the  2  ones,  making  12  ones.  Then  5  ones  from 
^T^    12  ones  leaves  7  ones. 

Having  taken  1  ten  from  the  3  tens,  but  2  tens  remain,  from 
which  we  cannot  take  7  tens.  Hence,  we  take  1  of  the  6  hundreds,  or 
10  tens,  and  add  it  to  the  2  tens,  making  12  tens.  Then  7  tens  from  12 
tens  leaves  5  tens. 


SUBTRACTION.  4'^ 

From  the  G  hundreds  we  have  already  taken  1  hundred,  leaving  5 
hundreds,  from  which  wc  now  take  3  hundreds. 

Queries. — Is  the  sum  of  the  subtrahend  and  the  remainder  equal  to 
the  minuend  ?  What  does  that  prove  ?  If  we  say  5  and  *t,  12  ;  7  and  1 
and  5,  13 ;  3  and  1  and  2,  6,  we  subtract  by  the  common  method^  of 
*'  making  change."    Can  you  explain  the  process  ? 

2.  From  543  take  29G. 
543  =^  5.00  +  40  +  3  =  500  +  30  +  13  =  400  +  130  +  13 


296  = 

200  +  90  +  6  = 

200  +  90  + 

6  =  200 

4-  90 

+  6 

247  = 

200 

+  40 

+  7 

Subtract  and 

prove  : 

3. 

4. 

6. 

6. 

7. 

8. 

9. 

211 

311 

773 

521 

626 

889 

342 

199 

279 

539 

234 

481 

691 

146 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

817 

824 

609 

508 

712 

793 

690 

677 

459 

368 

146 

299 

679 

664 

17. 

18. 

19. 

20. 

21. 

22. 

23. 

820 

924 

873 

856 

964 

903 

735 

698 

554 

297 

344 

299 

315 

285 

24. 

25. 

26. 

27. 

28. 

29. 

30. 

355 

871 

664 

124 

110 

485 

672 

179 

73 

493 

97 

20 

142 

210 

31. 

32. 

33. 

34. 

35. 

36. 

37. 

527 

956 

415 

133 

384 

694 

955 

329 

492 

116 

42 

115 

295 

61 

38. 

39. 

40. 

41. 

42. 

43. 

44. 

469 

783 

865 

614 

468 

821 

613 

252 

694 

572 

156 

79 

209 

108 

48 


SCHOOL  ARITHMETIC. 


Find  the  value  of  : 

45.  73251  -  23G79. 

46.  52175  -  37896. 

47.  04037  -  45069. 

48.  30524 


60. 

51. 
52. 


756328 
231562 
936061 

53.  238013 

54.  832415 


8765. 
49.  54321  -  12345. 

55.  Subtract  236  from  500. 
500  =  400  +  100  +  0  =  400  +  90  +  10 
236  =  200  +     30  +  6  =  200  +  30+6 
264  = 


467439. 
87968. 
847076. 
199815. 
243749. 


200 


60+4 


We  cannot  take  G  ones  from  0  ones,  neitlier  can  we  add  one  of  the 
tens,  for  there  are  no  tens.  Hence,  we  take  one  of  the  hundreds  and 
change  it  to  10  tens,  then  take  one  of  the  tens  and  change  it  to  10  ones, 
as  illustrated  in  tiie  operation. 

Then  6  ones  from  10  ones  leaves  4  ones  ;  3  tens  from  9  tens  leaves  6 
tens  ;  2  hundreds  from  4  hundreds  leaves  2  hundreds.  Hence  the  re- 
mainder is  264. 

Subtract  as  indicated  and  prove  : 


56. 

7250  - 

2894. 

68. 

1962057  - 

873698. 

57. 

5460  - 

3291. 

69. 

212.3456  - 

1745798. 

58. 

8300  - 

5867. 

70. 

7300892  - 

2006975. 

59. 

6500  - 

5678. 

71. 

5437001  - 

1008024. 

60. 

7000  - 

3269. 

72. 

8200345  - 

6832456. 

61. 

8000  - 

4753. 

73. 

1000001  - 

101002. 

62. 

6007  - 

2308. 

74 

6347543  - 

945876. 

63. 

12003  - 

7036. 

75. 

3702674  - 

2803879. 

64 

37200  - 

17501. 

76. 

9008007  - 

900809. 

65. 

17500  - 

2351. 

77. 

7685342  - 

6776387. 

66. 

91306  - 

4007. 

78. 

4433225  - 

3344668. 

67. 

10000  - 

6789. 

79. 

8076539  - 

8067845. 

Find  the  value  of  : 

80.  6792051  -  (139678  +  2005128)  -  1076053. 

81.  8076004  -  139678  +  2005128  -  1076053. 

82.  1000000  -  54321  -  (326000  +  240807). 


SUBTRACTION.  49 

83.  7236408  -  (6487543  ~  5328796)  -  4009007. 

84.  9001  +  (7200  -  6835)  -  (8003  -  3009)  -  4372. 

85.  59007  -  (32046  4-  3423)  -  (4008  +  13009  -  8729). 

86.  43700  +  56300  -  (3725  +  6275)  -  (90000  -  1234). 

87.  Add  one  million  to  the  difference  between  four  thou- 
sand seven  and  seven  thousand  four. 

88.  Subtract  the  difference  between  five  thousand  eighty- 
one  and  three  thousand  ninety-seven  from  the  difference 
between  four  thousand  nine  and  nine  thousand  four. 

89.  A  man  bought  a  farm  for  $17325  and  sold  it  for  $20000. 
How  much  did  he  gain  ? 

90.  How  long  is  it  since  the  discovery  of  America  by 
Columbus  in  1492  ? 

91.  A  man  owes  $2150,  but  has  only  $975.  How  much 
must  he  borrow  to  pay  the  debt  ? 

92.  Mont  Blanc  is  15572  feet  high,  and  is  3572  feet  higher 
than  Pike's  Peak.     What  is  the  height  of  the  latter  ? 

93.  Independence  was  declared  in  1776.  How  long  was 
that  after  the  discovery  of  America  ? 

94.  The  area  of  England  is  50922  square  miles,  and  that  of 
Pennsylvania  is  45215  square  miles.  How  much  larger  is 
England  than  Pennsylvania  ? 

95.  Mr.  B  having  $32700  gave  his  son  $10320,  and  his 
daughter  $8367.     I^ow  much  had  he  left  ? 

96.  In  1890  the  population  of  Pittsburg  was  238617,  and 
that  of  Philadelphia  was  1046964.  Which  had  the  larger 
population,  and  how  much  ? 

97.  A  train  left  Atlanta  with  273  passengers.  At  the 
first  station  24  got  off  and  9  got  on  ;  at  the  next  18  got  off 
and  12  got  on;  at  the  third  17  left  and  23  got  on  ;  at  the 
fourth  69  got  off.     How  many  yet  remained  on  the  train  ? 


98.  A,  B,  and  C  bought  a  store  for  $19325.     A  paid  $6105, 
and  B  paid  $753  more  than  A.     How  much  did  0  pay  ? 

99.  A  man  had  $10000  in  bank.     He  bought  a  house  for 

4 


50  SCHOOL  ARITHMETIC. 

$5G70,  paid  $1125  for  repairs,  137  for  insurance,  and  then 
sold  the  property  for  18500,  putting  the  money  in  bank. 
What  sum  did  he  then  have  in  bank  ? 

100.  Mr.  B  bought  two  horses,  paying  $120  for  one  and 
$145  for  the  other.  He  kept  them  six  weeks,  the  feed  for 
each  costing  him  $13,  and  then  sold  one  for  $170  and  the 
other  for  $210.     How  much  did  he  gain  ? 

101.  The  remainder  is  1786,  the  subtrahend  24G7.  AVhat 
is  the  minuend  ? 

102.  The  remainder  is  p,  the  subtrahend  q.  What  is  the 
minuend  ? 

103.  The  difference  between  two  numbers  is  796,  and  the 
larger  number  is  1275.     What  is  the  smaller  number  ? 

104.  The  difference  between  two  numbers  is  a,  and  the 
larger  number  is  h.     AVhat  is  tlie  smaller  number  ? 

105.  From  what  must  189  be  taken  to  leave  981  ? 

106.  From  what  must  a  be  taken  to  leave  h  ? 

107.  If  the  moon  is  240000  miles  from  the  earth,  and  the 
sun  92  million  miles,  how  much  farther  is  it  to  the  sun  than 
to  the  moon  ? 

108.  At  an  election  there  were  21635  votes  cast  for  A,  B, 
and  C.  A  got  9675,  B  got  327,  and  0  the  remainder.  How 
many  people  voted  for  C  ? 

109.  The  signs  +  and  —  were  used  by  Widman  in  an 
arithmetic  j^nblished  at  Leipzig  in  1489,  and  the  symbol  = 
by  Recorde  in  an  algebra  published  in  1557.  How  many 
years  elapsed  from  the  time  +  and  —  were  first  used  until 
the  =  was  used  by  Recorde  ? 

110.  The  greatest  depth  of  water  yet  measured  is  29400 
feet,  and  the  greatest  height  to  which  a  balloon  has  ascended 
is  37000  feet.  By  how  many  feet  does  the  greatest  height 
reached  exceed  the  greatest  depth  measured  ? 

111.  The  area  of  Virginia  is  42450  square  miles,  and  that 
of  West  Virginia  is  24780.  How  many  more  square  miles  has 
Virginia  than  West  Virginia  ? 


SUBTRACTION.  51 

112.  The  number  of  bushels  of  corn  raised  in  the  U.  S.  in 
1889  was  1,924,185,000,  and  the  number  of  bushels  of  wheat 
was  675,149,000.  How  many  more  bushels  of  corn  were 
raised  than  of  wheat  ? 

113.  In  1889  the  total  production  of  maple  sugar  in  the 
U.  S.  was  32,952,927  pounds.  In  1879  the  production  was 
36,576,061  pounds.     What  was  the  decrease  in  10  years  ? 

114.  The  total  oyster  product  of  the  U.  S.  in  1898  was 
25,349,668  bushels.  Maryland  produced  10,282,752  bushels 
and  Virginia  6,572,493  bushels.  How  many  bushels  did  all 
the  other  states  produce  ? 

Adding  and  subtracting  equal  numbers. — If  we  add 

10  to  each  side  of  the  equation 

12  +  8  -  9  =  25  -  16  +  2, 
we  have  12  +  8  -  9  +  10  =  25  -  16  +  2  +  10. 
Or,  subtracting  6  from  each  of  the  equals  in  the   same 
equation,  we  have 

12  +  8  -  9  -  6  =  25  -  16  +  2  -  6. 

1.  Does  adding  equal  numbers  to  equals  destroy  equality  ? 

2.  Does  subtracting  equal  numbers  from  equals  affect 
equality  ? 

67.  PRiisrciPLES. — 1,  If  equal  numbers  are  added  to  equals, 
the  sums  are  equal. 

2.  If  equal  numbers  are  subtracted  from  equals,  the  re- 
mainders are  equal. 


MULTIPLICATION. 

68.  1.  Suppose  you  have  a  roll  of  $5  bills  and  wish  to 
know  how  many  one-dollars  you  have.  How  can  you  find 
out  ? 

2.  Laying  the  bills  out  one  by  one,  you  may  count  by  5's, 
thus  :  5,  10,  15,  20,  25,  30.  In  doing  this  you  think  merely 
of  the  sum  of  the  addends,  and  not  of  the  number  oi  $5  bills. 

3.  Or  you  may  first  count  the  bills — 1,  2,  3,  4,  5,  6.  Six 
bills,  each  %o.  How  many  times  is  15  repeated  to  make  your 
$30.     Did  you  tliink  of  this  six  in  the  former  process  ? 

Observe  that  this  process  introdnces  the  idea  of  times — an 
idea  not  present  in  addition. 

4.  Your  money  is  measured  in  two  ways,  by  15  and  by  11. 
When  the  unit  of  measure  is  $5,  how  many  times  is  it  re- 
peated ?     How  many  when  the  unit  is  II? 

5.  It  should  be  remarked  that  this  higher  process  is  founded 
upon  addition.  AVe  learn  that  six  5's  are  30  (ones)  by  first 
finding  the  sum,  and  then  noticing  hem  many  times  the  5  is 
repeated  to  make  that  sum. 

6.  Six  times  2  quarts  are  how  many  '^  one-qnarts^'  ?  How 
many '^  six-quarts  ^^  ? 

7.  A  man  measuring  his  corn  filled  a  2-peck  measure  8 
times  ;  how  much  corn  had  he  ? 

8.  In  the  last  example  what  is  the  unit  of  measure  ?  How 
many  times  was  it  taken  ?  What  quantity  did  he  measure  ? 
Did  he  wish  to  find  how  many  ^^ two-pecks"^  he  had,  or  how 
many  pechs  9 

9.  Ella  bought  10  yards  of  cloth  at  $2  a  yard.  How  much 
did  it  cost  her  ? 


MtJLTlPLiCAtioiJ.  53 

How  many  times  is  $2  repeated  or  taken  to  make  $20  ? 
What  is  here  used  as  a  unit  of  measure  ?  Is  |2  itself  a  meas- 
ured quantity  ?     By  what  unit  is  it  measured  ? 

The  cost  is  10  x  $2 — ten  units  of  $2  each— but  by  the  process  of  mul- 
tiplication we  change  this  to  $20,  that  is,  to  20  units  of  $1  each. 

09.  The  process  of  taking  one  number  as  many  times  as 
there  are  units  in  another  is  called  Multiplication. 

70.  The  number  multiplied  by  another — the  number  taken 
so  many  times — is  called  the  Multiplicand.  It  is  regarded 
as  a  unit  of  measure. 

71.  The  number  that  tells  how  many  times  the  multipli- 
cand is  taken  is  called  the  Multiplier.     It  is  pure  number. 

72.  The  result  obtained  by  multiplying  is  called  the 
Product.     It  is  regarded  as  measured  quantity. 

73.  The  multiplicand  and  multiplier  are  the  Factors  of 
the  product. 

74.  The  product  of  two  numbers  is  the  same  whichever  is 
taken  as  the  multiplier. 

Of  course  it  is  absurd  to  say  that  3  times  15  is  the  same  as 
$5  times  3  ;  but  the  meaning  is  that 

3xl5  =  3x5x$l=:5x3x$lrrr5x|3.     • 

75.  The  Sign  of  Multiplication  (  x )  is  read  times  when 
it  follows  tlie  multiplier,  and  multiplied  by  when  it  precedes 
the  multiplier. 

Thus,  3  X  $5  is  read  3  times  $5  ;  $5  x  3  is  read  $5  multiplied  by  3. 
This  symbol  was  first  used  by  Oughtred  in  1631. 

1.  When  we  multiply  $5  by  3,  is  the  multiplicand  concrete 
or  abstract  ?     Is  the  multiplier  concrete  ?     Is  the  product  ? 

2.  Can  we  multiply  U  by  13  ?  4  by  $3  ?  $5  by  2  feet  ?  By 
what  can  we  multiply  15  ? 


u 


SCHOOL  ARITHMETIC. 


76.  Prin"Ciples. — 1.   The  multiplier  is  ahmys  an  abstract 
number. 

2.   The  product  is  always  like  the  multiplicand. 


MULTIPLICATION    TABLE. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 
11 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 
132 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

This  table  may  be  used  in  two  ways.     Find  out  the  two  ways. 

77.  The  following  exercises  should  be  practiced  daily  until 
the  pupil  is  able  to  announce  at  sight  the  product  of  any  two 
numbers  from  2  to  12. 

Announce  products  at  sight  : 


1.  5  X  7 

4x9 

2  X  11 

7x2 

11  X  4 

2.  9  X  5 

3x6 

3x9 

6  X  10 

7x8 

3.  5  X  11 

7  X  10 

3  X  12 

7x4 

2x9 

4.  4  X  4 

6x3 

8x6 

10  X  5 

9x8 

5.  5  X  3 

4x5 

4  X  12 

7x6 

11  X  11 

6.  7  X  9 

5x8 

6x5 

8x7 

4x6 

7.  5  X  6 

3x8 

10  X  2 

5x9 

11  X  8 

MULTIPLICATION. 


55 


8.  0  X 

4 

3 

X 

7 

6 

X 

12 

8 

X 

10 

9x7 

9.  5  X 

5 

12 

X 

3 

8 

X 

8 

7 

X 

5 

9x9 

10.  6  X 

6 

7 

X 

12 

10 

X 

11 

12 

X 

5 

10  X  4 

11.  9  X 

4 

10 

X 

3 

8 

X 

4 

4 

X 

11 

3  X  11 

12.  4  X 

10 

12 

X 

11 

10 

X 

7 

7 

X 

3 

11  X  6 

13.  9  X 

2 

8 

X 

3 

10 

X 

6 

6 

X 

8 

12  X  12 

14.  4  X 

8 

10 

X 

10 

8 

X 

8 

5 

X 

10 

8  X  11 

15.  7  X 

7 

12 

X 

9 

4 

X 

7 

8 

X 

12 

10  X  9 

16.  9  X 

12 

11 

X 

3 

12 

X 

8 

9 

X 

10 

12  X  10 

17.  6  X 

11 

2 

X 

12 

11 

X 

9 

12 

X 

4 

6x9 

18.  5  X 

12 

11 

X 

10 

10 

X 

8 

9 

X 

11 

6x7 

19.  8  X 

5 

12 

X 

7 

7 

X 

11 

5 

X 

4 

12  x  2 

20.  5  X 

11 

11 

X 

2 

12 

X 

G 

10 

X 

12 

11  X  12 

For  further  practice 
in  rapid  work  place 
these  diagrams  on  the  ^ 
blackboard,  and  then, 
.  using  the  numbers  at 
the  centers  as  multi- 
pliers, indicate  the  7 
multiplicands  with  the 
pointer. 

ORAL    EXERCISES. 


A 


V 


78.  1.  At  $4  a  day,  how  much  can  I  earn  in  6  days  ? 

2.  There  are  12  things  in  a  dozen.     IIow  many  things  are 
in  8  dozen  ? 

3.  How  much  will  9  yards  of  calico  cost  at  6  cents  a  yard  ? 

4.  How  far  does  a  man  walk  in  10  hours  at  the  rate  of  4 
miles  an  hour  ? 

5.  At  $6  each,  how  much  must  he  paid  for  12  sheep  ? 

6.  Ben  picked  9  quarts  of   cherries,  and  Fred  picked  8 
times  as  many.     IIow  many  quarts  did  Fred  pick  ? 

7.  If  my  hens  lay  7  eggs  each  day,  how  many  will  they 
lay  in  9  days  ? 

8.  There  are  8  quarts  in  a  peck.     How  many  quarts  in  7 
pecks  ?    In  a  bushel  ? 


56  SCHOOL  ARITHMETIC. 

4 

9.  Frank  went  to  the  post  office  8  days,  and  each  time  got 
no  letters.  How  many  letters  did  he  get  in  the  8  days  ?  Then 
8x0  =  what  ? 

10.  A  man  bought  a  dozen  slates  at  10  cents  each.  How 
much  change  did  he  get  out  of  a  two-dollar  bill  ? 

11.  Find  the  cost  of  8  yards  of  velvet  at  $5  a  yard. 

(a).  Since  one  yard  costs  $5,  8  yards  cost  8  times  $5,  or  $40. 

(b).  At  $1  a  yard  8  yards  would  cost  $8  ;  hence,  at  $5  a  yard,  the 
cost  is  5  times  $8,  or  $40. 

Note. — The  pupil  should  compare  these  solutions  and  make  himself 
thoroughly  familiar  with  the  process  in  each. 

12.  James  bought  15  papers  at  3  cents  each  and  sold  them 
for  5  cents  each.     How  much  did  he  make  ? 

13.  Dick  earns  $23  a  week,  and  Charles  earns  $10  a  week. 
In  6  weeks  Dick  earns  how  much  more  than  Charles  ? 

14.  Two  men  start  from  the  same  place,  one  going  west  at 
the  rate  of  7  miles  an  hour,  the  other  going  east  at  the  rate 
of  4  miles  an  hour.     How  far  apart  are  they  in  4  hours  ? 

16.  John  has  twice  as  many  apples  as  Mark.  How  many 
have  both,  if  Mark  has  2  times  8  apples  ? 

16.  A  and  B  start  from  Boston  and  travel  in  the  same 
direction,  A  going  25  miles  an  hour,  and  B  11  miles.  How 
far  are  they  apart  in  3  hours  ? 

17.  A  has  $4,  B  has  3  times  as  much,  and  C  has  twice  as 
much  as  B.     How  much  have  they  all  ? 

18.  How  much  will  5  pencils  cost  at  h  cents  each  ? 

19.  If  a  man  walks  a  miles  an  hour,  how  far  will  he  walk 
in  h  hours  ? 

20.  A  man  having  12a  dollars  bought  5  sheep  at  2«  dollars 
each.     How  much  had  he  left  ? 

21.  If  a  man  spends  h  dollars  a  day  for  c  days,  how  much 
does  he  spend  ? 

22.  A  man  having  a  dollars  paid  p  dollars  each  for  q  cows. 
How  much  had  he  left  ? 

23.  Harry  has  jt?  marbles,  Ira  Jias  q  marbles,  and  Sam  has 
r  times  as  many  as  both  together.     How  many  has  Sam  ? 


MULTIPLICATION. 


57 


24.  When  hats  are  worth  $5  each,  how  much  are  3  hats 
worth  ?     How  do  you  know  ? 

26.  Why  do  3  hats  cost  3  times  $5  ? 

26.  Complete  the  following  :  When  hats  are  worth  a  dollars 

each,  7  hats  are  worth dollars,  because  7  hats  are  worth 

as  mucli  as . 


79.  Announce  products  at  sight : 


G. 

2. 

2. 

3. 
X  2. 
X  3. 
X  11. 
X  12. 


9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 


12. 

4. 

3. 

2. 

3. 

2. 

3. 


17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 


11. 

12. 

20. 

3. 

2. 

3. 

11. 

2. 


WRITTEN     EXERCISES. 


80.  1.  Multiply  365  by  4. 


365  =    300 
4  = 


60  + 


1460  =  1200  +  240  +  20 
or  1200  +  260  +  0 
or  1400  +60+0 


(b) 

365 

365 

365 

365 

1460 


(a)  (b)  4  times  5  ones  are  20  ones, 

or  2  tens  and  no  ones.  We 
write  the  0  in  ones'  place  in  the 
product,  and  add  the  2  tens  to 
4  times  6  tens,  making  26  tens, 
or  2  hundreds  and  6  tons.  We 
write  the  6  in   tens'   place  in 

the  product,  and  add  the  2  hundreds  to  4  times  3  hundreds,  making  14 

hundreds,  or  1  thousand  and  4  hundreds. 

How  is  the  same  result  obtained  in  (b)  ?    Why  do  we  write  the  multi, 

plier  under  the  multiplicand  ?    Find  by  trial  if  you  can  multiply,  begia 

ning  at  the  left. 

Multiply  the  following : 

2.  253  by  3.  8.  697  by  8.  14.  287  by  9. 

3.  426  by  3.  9.  853  by  9.  15.  913  by  8. 

4.  736  by  4.  10.  437  by  4.  16.  368  by  7. 

5.  594  by  6.  11.  978  by  5.  17.  627  by  6. 

6.  8.67  by  5.  12.  634  by  6.     *  18.  598  by  5. 

7.  539  by  7.  .  13.  549  by  7.  19.  817  by  9. 


58  SCHOOL   ARITHMETIC. 

20.  62,  417,  685  by  4.  26.  98,  765,  432  by  6. 

21.  85,  236,  417  by  6.  27.  24,  681,  357  by  9. 

22.  47,  395,  173  by  5.  28.  36,  925,  814  by  8. 

23.  18,  236,  456  by  7.  29.  62,  847,  418  by  7. 

24.  54,  923,  687  by  9.  30.  56,  847,  389  by  5. 

25.  37,  281,  459  by  8.  31.  12,  345,  679  by  9. 

81.  In  expressions  like  4  +  2x3,  the  operation  indicated 
by  the  sign  x  must  be  first  performed. 

Thus,  4+3x3  means  4  +  6,  not  6  x  3.  18  -  8  x  2  means  18  -  16, 
not  10  X  2  ;  but  (18  -  8)  x  2  means  10  x  2.  (4  +  2)  x  3  =  18  ;  but 
4  +  2  X  3  =  10.  Of  course  4  +  (2  x  3)  =r  10,  but  when  two  or  more  num- 
bers are  connected  by  the  sign  x  the  parentheses  are  superfluous. 

Find  the  value  of  : 

20  -  9  X  2  +  3  X  5  -  7. 
25  +  4  X  3  -  6  x  6  +  9. 
12  X  4  -  3  X  7  +  10  X  5. 
12  X  (4  -  3)  X  7  +  10  X  5. 
36  +  175  X  4  -  89  X  6  -  200. 
1000  -  97  X  8  +  1  -  3  X  75. 

82.  To  multiply  by  lO,  100,  lOOO,  etc. 

1.  When  a  number  is  multiplied  by  10,  what  is  annexed 
to  it  ? 

Then  what  is  the  shortest  method  of  finding  10  times  any 
number  ? 

2.  AVhen  a  number  is  multiplied  by  100,  what  is  annexed 
to  it? 

Then  what  is  the  shortest  method  of  finding  100  times 
any  number  ? 

3.  When  the  multiplier  is  10,  100,  1000,  etc.,  how  many 
ciphers  must  be  annexed  to  the  multiplicand  ?  Find  out  by 
trial. 

83.  Principle. — A  number  is  multiplied  hy  10,  100, 1000, 
etc.,  hy  annexing  m  many  ciphers  to  the  multiplicand  as  there 
are  ciphers  in  the  multiplier. 


1. 

6  +  4x2. 

7. 

2. 

7  +  3x6. 

8. 

3. 

2  +  8x3. 

9. 

4. 

8-3x2. 

10. 

5. 

16  -  5  X  3. 

11. 

6. 

12  +  4  X  7. 

12. 

MULTIPLICATION.  59 

WRITTEN     EXERCISES. 

84.  Copy,  and  complete  the  equations  : 

1.  25.xl0=:(     ).       37xlOO  =  ( 

2.  50xlO=(     ).     175xl00=( 

3.  100  X  10  ==  (     ).     387  X  100  =  ( 

4.  41G  X  10  =  (     ).     nOO  X  100  =  ( 

5.  708  X  10=(     ).     914  X  100=( 

6.  Multiply  743  by  5000. 
743000        5000  is  5  times  1000.      ITence  we  first  annex  3  ciphers, 
5     which  multiplies  743  by  1000,  aiul  then  multiply  by  5. 


59xl000  =  (     ). 

324  X  1000  =  (     ). 

872  X  1000  =(    ). 

903  X  1000  =  (    ). 

1000  X  1000  =  (     ). 

T^ouKi  we  luuiuiniy  nrst  oy  0 
3715000                              ^  ^            ^ 

ana  tnen  oy  luuu  r 

Find  the  products  of  : 

7.  578  X  30.            12.   924  x  700. 

17.  8125  by  6000. 

8.  G39  X  20.             13.  538  x  400. 

18.  5346  by  8000. 

9.  825  X  40.             14.   190  x  GOO. 

19.  4325  by  9000. 

10.  347  X  50.             15.  467  x  800. 

20.  6913  by  7000. 

11.   718  X  GO.             16.  28G  x  900. 

21.  9000  by  6000. 

85.  To  fiiKl  the  product  when  the  right-hand  figure 

of  the  multiplier  is  significant ;  that  is,  is  not  0. 

1.  Multiply  327  by  245. 

327 

327 

245  =  200  +  40  +  5 

245 

1635  =z      5  X  327 

1635 

13080  =    40  X  327 

1308 

65400  =  200  X  327 

654 

80115  =  245  X  327  80115 

The  multiplier  245  =  200  +  40  +  5.  Hence  we  multiply  first  by  5, 
then  by  40,  then  by  200,  and  then  add  the  partial  products. 

Or,  since  40  is  4  tens,  we  may  multiply  by  4  tens  instead  of  40.  The 
product  is  1308  tens,  hence  the  8  must  be  written  under  the  tens,  and  the 
ones'  place  left  vacant.  Since  200  is  2  hundreds,  we  may  multiply  by  2 
hundreds  instead  of  200.  The  product  is  654  hundreds,  hence  the  right- 
hand  figure  must  be  written  under  hundreds,  leaving  vacant  two  places 
to  the  right. 


60 


SCHOOL  ARlTttMETiC. 


DiR^CTto^.-^  Write  the  multiplier  under  the  multiplicand, 
ones  under  ones,  tens  under  tens,  etc.  Multiply  the  multipli- 
cand first  hy  the  ones  of  the  mMltiplier,  then  by  the  tens,  and 
so  on,  placing  the  right-hand  figure  of  each  product  directly 
under  the  figure  of  the  multiplier  used  to  obtain  it ;  then  add 
the  several  products  thus  obtained. 

To  insure  accuracy,  review  the  work  carefully,  or  multiply 
the  multiplier  by  the  multiplicand,  and  compare  results. 


WRITTEN 

EXERCISES 

Find  the  product  of  : 

2.  375 

X  24.      11.  943 

X 

235. 

20.  8076  X  607. 

3.  628 

X  35.      12.  627 

X 

364. 

21.  4109  X  385. 

4.  296 

X  16.      13.  548 

X 

621. 

22.  7340  X  468. 

5.  418 

X  29.      14.  703 

X 

513. 

23.  5897  X  903. 

6.  537 

X  32.      15.  864 

X 

207. 

24.  8645  X  795. 

7.  704 

X  58.      16.  597 

X 

648. 

25.  9857  X  606. 

8.  650 

X  43.      17.  486 

X 

753. 

26.  7469  X  729. 

9.  486 

X  74.      18.  398 

X 

462. 

27.  7208  X  905. 

10.  825 

X  86.      19.  987 

X 

789. 

28.  8900  X  547. 

Multiply  as  indicated  : 

29. 

37  X  425  X  8000. 

35. 

456 

X  723  X  800. 

30. 

72  X  618  X  6000. 

36. 

925 

X  860  X  400. 

31. 

43  X  703  X  5000. 

37. 

378 

X  409  X  700. 

32. 

40  X  900  X  7823. 

38. 

846 

X  520  X  290. 

33. 

83  X  750  X  8600. 

39. 

750 

X  693  X  804. 

34. 

69  X  583  k  4009. 

40. 

623 

X  954  X  287. 

41. 

421896  X  215. 

48. 

516057  X  579. 

42. 

373842  X  327. 

49. 

439104  X  806. 

43. 

654083  X  456. 

50. 

182564  X  975. 

44. 

937584  X  274. 

51. 

869375  X  658. 

45. 

160835  X  621. 

52. 

82047  X  4306. 

46. 

395827  X  768. 

53. 

47638  X  5219. 

47. 

284936  X  492. 

54. 

68049  X  3168. 

MULTIPLICATION.  Ql 

65.  37583  x  6574.  69.  18254  x  9651. 

66.  38065  X  4018.  60.  75864  x  3972. 

67.  92736  x  8539.  61.  68357  x  4098. 

68.  72069  X  3964.  62.  37528  x  6573. 

Find  the  value  of  : 

63.  (703  +  815)  X  (5403  -  765)  -  860  x  1354. 

64.  876  X  234  -  98  +  56  x  1009  -  12895  x  8. 

65.  (345721  -  18693)  x  8700  -  63594  x  27. 

66.  (5389  -  76  X  59)  x  86  -  (51839  +  11  x  1987). 

67.  (5678  -  79)  x  84  x  68  -  52389  -  8760  x.  937. 

68.  62007  X  503  -  8460  x  790  -  37625  x  89. 

69.  4587  +  7854  x  79000  -  78  x  (673005  +  378900). 

70.  (99  -  88)  X  180  4-  10000  -  77  x  66  -  179  x  38. 

71.  I  paid  $2678  for  426  bushels  of  clover  seed,  which  I 
sold  at  S7  a  bushel.     How  much  did  1  gain  ? 

72.  John  drives  the  cows  home  to  be  milked  every  morn- 
ing and  evening.  If  the  pasture  is  80  rods  from  home,  how 
far  does  John  travel  in  4  weeks  ? 

73.  Two  trains  left  Chicago  at  the  same  time,  one  going 
west  at  the  rate  of  36  miles  an  hour,  the  other  east  at  the 
rate  of  28  miles  an  hour.  How  far  apart  were  they  in  17 
hours  ? 

74.  A  has  $95,  B  has  three  times  as  much,  and  C  has  twice 
as  much  as  both.  How  much  more  than  $1000  have  they 
all? 

76.  There  are  5280  feet  in  a  mile.  If  steel  rails  weigh  36 
pounds  to  the  foot,  what  is  the  weight  of  the  rails  in -two 
miles  of  double-track  railway  ? 

76.  If  a  cow  is  worth  7  sheep,  and  a  horse  is  worth  6  cows, 
how  many  sheep  are  worth  as  much  as  19  horses  and  28  cows  ? 

77.  A  boy  who  lives  167  rods  from  the  schoolhouse  goes 
to  school  regularly.  If  he  goes  home  for  dinner  every  day, 
how  far  does  he  walk  in  a  week  ? 

78.  A  drover  bought  286  sheep  for  $1430.     After  feeding 


62  SCHOOL  AKITHMETIC. 

them  a  week  at  an  expense  of  $175,  he  sold  97  of  them  at  $8 
each,  and  10  of  them  died.  He  sold  the  remainder  for  $900. 
How  much  did  he  gain  on  all  ? 


79.  A  lady  bought  15  yards  of  calico  at  6  cents  a  yard,  3 
yards  of  velvet  at  $1.49  a  yard,  and  27  yards  of  muslin  at  <S 
cents  a  yard.  She  gave  the  clerk  a  ten-dollar  bill.  How 
much  change  should  she  receive  ? 

80.  H  the  bill  just  mentioned  was  counterfeit,  how  much 
did  the  store  lose  ? 

81.  Mr.  A  bought  17  five-dollar  bills  at  2  pigs  each.  How 
many  pigs  did  be  pay  for  them  ? 

82.  Since  the  product  is  the  sum  of  two  or  more  equal 
numbers,  one  of  which  is  the  multiplicand,  how  may  the 
multiplier  be  found  when  the  product  and  multiplicand  are 
given  ? 

83.  When  the  product  is  30,  how  often  can  the  multipli- 
cand 4  be  subtracted  from  the  product  ?  Then  what  is  the 
multiplier  ? 

84.  The  product  is  90,  the  multiplicand  6.  What  is  the 
multiplier  ? 

85.  The  multiplicand  is  18a,  the  product  54f«.  What  is 
the  multiplier  ? 

86.  When  the  product  is  456  and  the  multiplicand  5b, 
what  is  the  multiplier  ? 

87.  How  many  groups  of  32^;  =  W'Zp? 

86.  1.  If  a  rectangle  4  ft.  long  and  2  ft.  wide  is  divided 
as  in  the  figure,  what  will  each  of  the 
small  squares  represent  ?  What  is  the 
area  of  one  of  the  strips  ?  (4x1  sq.  ft.). 
Then  what  is  the  area  of  the  two  strips, 
or  of  the  rectangle  ? 

2.  How  many  square  feet  in  the  surface  of  a  table  5  ft. 
long  and  3  ft.  wide  ? 


B 


MULTIPLICATION. 


63 


3.  What  is  the  area  of  a  rectangular  field  40  rods  long  and 
35  rods  wide  ? 

4.  IIow  many  square  feet  of  carpet  will  cover  a  room   16 
feet  square  ? 

6.  A  rectangle  is  8  feet  long  and  6  feet  wide.     What  is  its 
area  ? 

6.  What  is  the  area  of  a  rectangle  a  feet  long  and  h  feet 
wide  ? 

7.  IIow  many  square  feet  of  oilcloth  will  cover  a  floor  b  feet 
square  ? 

8.  The  area  of  a  rectangle  5  feet  long  is  20  square  feet. 
How  wide  is  it  ? 

9.  How  many  square  miles  in  a  tract  of  land  300  miles  long 
and  half  as  wide  ? 

10.  In  this  figure 
let  AB  =  10  ft.,  BC 
=  8  ft.,  DE  =  5  ft., 
EF  =  7  ft.,  FG  =  9 
ft.,  GH  =  5  ft. 

(a)  Find  the 
length  of  DC  and 
of  AH. 

(b)  Divide  the  fig- 
ure into  rectangles 
and  find  its  area. 

11.  The  top  of  a  desk  is  a  rectangle,  5  feet  long  and  3  feet 
wide.     What  is  its  area  in  square  feet  ? 

12.  How  much  available  storage  space  is  in  a  building  300 
feet  long,  150  feet  wide,  and  six  stories  high  ? 


G 

D 

F 

E 

n 


DIVISION. 

87.  1.  What  number  multiplied  by  3  gives  115  as  a  prod- 
uct 

2.  By  what  must  6^'  be  multiplied  to  give  18^'  as  a 
product  ? 

3.  If  a  man  earus  $15  in  3  days,  how  much  is  that  a  day  ? 
3  X  ($ )  =  $15. 

4.  Product  42,  one  factor  7  ;  what  is  the  other  factor  ? 

5.  In  multiplication  two  factors  are  given  to  find  the 
product.  In  division  the  product  and  one  factor  are  given. 
What  is  to  be  found  ? 

88.  The  process  of  finding  either  factor  when  the  other 
factor  and  their  product  are  given  is  called  Division. 

89.  One  of  the  two  factors  is  one  of  the  equal  parts  of  a 

quantity  or  number,  the  other  being  the  number  of  equal 
parts.     Hence  division  includes — 

(A).  Finding  how  many  equal  parts  of  a  given  size  com- 
pose a  number. 

This  involves  the  idea  of  measuring,  being  contained 
in,  as  seen  in  the  following  exercises  : 

1.  Divide  a  12-foot  line  into  equal  parts,  each  3  feet  in 
length.  How  many  equal  parts  are  there  ?  How  many  3's 
in  12  ? 

2.  There  are  10  apples  on  a  plate.  How  many  times  can 
you  take  2  apples  away  ?     How  many  2^s  in  10  ? 

3.  How  many  groups  of  4  books  each  can  be  made  from  24 
books  ?    How  many  4's  in  24  ? 


DIVISION.  g5 

4.  How  many  five-dollar  bills  make  $20,  or  how  many 
times  is  $5  contained  in  $20  ? 

5.  A  man  has  $40  in  packages  of  $5  each.  How  many 
packages  has  he  ?  Of  how  many  equal  parts  is  his  $40  made 
up,  or  composed  ? 

6.  How  many  times  must  we  take  $5  to  make  $40  ? 

8  X  $5  =  $40  ;  then  $40  -j-  $5  = .     That  is,   $5  is  contained  8 

times  in  $40. 

(B).  Finding  one  of  the  equal  parts  of  a  number. 

This  involves  the  idea  of  separation,  partition,  as  may 

be  seen  in  the  following  exercises  : 

1.  Divide  G  feet  into  2  equal  parts.  AVhat  is  the  length  of 
each  part  ? 

2.  What  is  one  of  the  4  equal  parts  of  $12  ?     $12  -f-  4  = 


3.  What  must  be  multiplied  by  3  to  give  $18  as  a  product  ? 
$18  -^  3  =  $ . 

4.  What  is  the  weight  of  each  package  when  20  lb.  of 
butter  is  divided  into  4  equal  parts  ?     20  lb.  -f-  4  = lb. 

6.  When  $40  is  divided  equally  among  8  boys,  how  much 
does  each  boy  get  ?  What  is  one  of  the  8  equal  parts  of 
$40  ?     $40  -=-  8  =  $ . 

90.  When  the  product  and  given  factor  are  like  numbers, 
we  have  the  process  mentioned  in  (A);  that  mentioned  in 
(B)  applies  when  they  are  unlike  numbers. 

91.  The  given  factor  shows  either  the  number  of  equal 
parts  or  the  size  of  them,  and  is  called  the  Divisor. 

92.  The  factor  to  be  found  is  called  the  Quotient. 

93.  The  given  product  is  called  the  Dividend.     It  is  the 

sum  of  the  equal  parts. 

1.  When  books  are  $2  apiece,  how  many  can  you  buy  for 
|9  ?     How  much  money  will  you  have  left  ? 
5 


66  SCHOOL  ARITHMETIC. 

2.  How  many  3^s  in  9  ?  How  many  in  11  ?  How  many 
left  over  ? 

3.  When  yoii  take  three  5's  from  17,  how  many  remain  ? 

4.  When  you  take  5  cents  from  17  cents  three  times,  how 
many  cents  remain  ?  Is  the  remainder  a  part  of  the  17 
cents  ?     Is  it  greater  or  less  than  the  5  cents  ? 

94.  The  number  left  over,  or  remaining,  when  the  division 
is  not  exact  is  called  the  Remainder.  It  is  a  part  of  the 
dividend,  and  always  less  than  the  divisor. 

95.  The  Hign  of  Division  (-^-)  is  read  diinded  hy,  and 
when  i^laced  between  two  numbers  shows  that  the  one  on  the 
left  is  to  be  divided  by  the  one  on  the  right. 

Thus,  24  -i-  3  is  read  twenty-four  divided  hy  three.  Division  may  also 
be  indicated  by  writing  the  divisor  under  the  dividend,  thus,  V^. 

This  sign  of  division  (-^)  was  first  used  by  Rahn  in  an  algebra  pub- 
lished at  Zurich  in  1659. 

96.  The  product  of  divisor  and  quotient,  plus  the  re- 
mainder, if  any,  is  equal  to  the  dividend. 

1.  5  X  $4  =  how  many  dollars  ? 

2.  How  many  times  $4  =  $20  ?     $20  ^H=  what  ? 

3.  In  example  1,  what  is  the  product  ?  What  is  the 
dividend  in  example  3  ? 

4.  Name  the  multiplicand  in  example  1.  In  example  2, 
what  is  the  divisor  ? 

97.  Division  is  the  converse  of  multiplication.  Dividend 
corresponds  to  product,  divisor  to  one  factor,  and  quotient  to 
the  other.  Hence  to  find  a  quotient  we  need  only  recall  how 
many  of  the  divisors  equal  the  dividend. 

Thus,  to  finxl  the  quotient  of  18  -4-  3,  we  recall  the  fact  that  six  3's 
are  18  ;  hence  we  know  that  6  is  the  quotient. 


DIVISION. 


67 


98.  Name 

1.  24 -f- 4. 

2.  45-4-5. 

3.  36  -h  6. 

4.  35  -^  7. 

5.  60-^5. 

6.  45  -4-  9. 

7.  32  -4-  8. 

8.  42  --  7. 

9.  32  -V-  4. 

10.  40 -v^  4. 

11.  48-^6. 

12.  36  ~  9. 

13.  56  -^  7. 

14.  60  -^  6. 

15.  64 --8. 


quotients  at  sight  : 


16.  35 

17.  3G 

18.  40 

19.  54 

20.  63 

21.  81 

22.  42 

23.  48 

24.  63 

25.  72 

26.  70 

27.  66 

28.  72 

29.  44 

30.  80 


^5. 
-4-4. 

4-7. 
-h  6. 
-4-9. 
-4-9. 

-4-  6. 

-^8. 

-4-  7. 
-4-  9. 

^7. 

-4-  6. 
-4-  6. 

-f-4. 


31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 
45. 


28-4-  4. 

30  -4-  6. 
77--    7. 

40 -^  8. 

36^  3. 

27^  9. 

56^  8. 

72 -J-  8. 

88-4-  8. 

90 -f-  9. 

84-4-  7. 

99-4-  9. 
90  -4-  10. 

28^  7. 

27 -f-  3. 


Name  quotients  and 
61.   6)44       71.  8)83 


62.  4)^ 

63.  9)^ 

64.  8)^ 

65.  5)38 

66.  8)49 

67.  9)29 

68.  7)65 

69.  6)63 

70.  7)72 


72.  4)JU) 

73.  7)_75 

74.  9)6() 

75.  8)J30 

76.  9)48 

77.  8)52 

78.  3)^ 

79.  4)47 

80.  5)58 


•emainders 

81.  6)51 

82.  4)22 

83.  6)^ 

84.  9)38 

85.  8)^ 

86.  9)^ 

87.  5)j63 

88.  6)^ 

89.  3)^ 

90.  8)99 


91. 
92. 
93. 
94. 
95. 
96. 
97. 
98. 
99. 
100. 


46.  48-4-    4. 

47.  60 -^    5. 

48.  96-4-    8. 

49.  108-4-    9. 

50.  no -4- 10. 

51.  72  -4-  12. 

52.  121  -4-  11. 

53.  48  -4-  12. 

54.  36  -4-    3. 

55.  60  -J-  12. 

56.  132-4-11. 

57.  80  ^  10. 

58.  120-4-12. 

59.  144  -4-  12. 

60.  132  -4-  12. 


4)51 

7)89 

9)^ 

8)^ 

9)53 

7)^ 

8)61 

9)103 

8)101 

6)79 


101.  8)^ 

102.  11)105 

103.  12)107 

104.  12)117 

105.  10)126 

106.  12)149 

107.  11)138 

108.  9)86 

109.  7)^ 

110.  12)141 


ORAL   EXERCISES. 


99.  The  first  13  examples  following  involve  applications 
of  the  process  mentioned  in  (A).  The  quotient  in  each  is 
abstract;  it  corresponds  to  the  multiplier. 


m  SCHOOL  ARlTHMElTld. 

1.  When  apples  are  13  a  barrel,  how  many  barrels  can  be 
bought  for  $21 ? 

At  $3  a  barrel,  as  many  barrels  can  be  bought  for  $21  as  $3  is  con- 
tained times  in  |21.  $3  is  contained  7  times  in  $21.  Hence  7  barrels 
can  be  bought. 

It  will  be  observed  that  in  the  division  the  quotient  7  is  not  7  barrels, 
but  is  an  abstract  number,  which  is  interpreted  or  applied  in  the  co7i- 
clusion. 

2.  At  5  cents  each,  how  many  tops  can  be  bought  for  40 
cents  ? 

3.  There  are  4  quarts  in  one  gallon.  How  many  gallons  in 
32  quarts  ? 

4.  If  10  yards  of  cloth  make  a  dress,  how  many  dresses  can 
be  made  from  90  yards  ? 

5.  A  farmer  put  3G  bushels  of  wheat  into  three-bushel 
bags.     How  many  bags  did  he  fill  ? 

6.  In  an  orchard  are  72  apple  trees.  If  there  are  12  trees 
in  a  row,  how  many  rows  are  there  ? 

7.  A  gardener  tied  48  onions  in  bunches  of  6  each.  How 
many  bunches  did  he  make  ? 

8.  Mr.  B  gave  $56  to  some  boys.  If  each  boy  got  $8,  to 
how  many  boys  did  he  give  money  ? 

9.  How  many  weeks  in  63  days  ? 

10.  Mrs.  A  has  54  chickens  in  coops.  If  she  has  9  chick- 
ens in  a  coop,  how  many  coops  has  she  ? 

11.  I  paid  $56  for  coal  at  $7  a  ton.    Howmanytonsdid  Iget? 

12.  How  many  slates  at  8  cents  each  can  be  bought  for 
72  cents  ? 

13.  A  butcher  invested  $84  in  fat  pigs,  paying  $12  for  each. 
How  many  did  he  get  ? 

14.  What  number  multiplied  by  7  equals  56  ? 

15.  What  number  divided  by  12  gives  the  quotient  6  ? 

16.  How  many  times  can  6  be  subtracted  from  54  ? 

17.  How  many  apples  at  a  cents  each  can  be  bought  for 
5a  cents  ? 


DIVISION. 


69 


18.  A  man  spent  18b  dollars  for  books  at  3b  dollars  apiece. 
How  many  did  he  get  ? 

19.  What  number  divided  by  7a  gives  the  quotient  4  ? 

20.  How  many  two-cent  stamps  can  be  bought  for  a  cent 
and  a  quarter  ? 

lOO.  One  of  the  two  equal  parts  of  a  number  is  one-half 
of  it,  one  of  the  three  equal  parts,  one-third,  one  of  the  four 
equal  parts,  one-fourth,  three  of  the  four  equal  parts,  three- 
fourths,  and  so  on. 


One-half  is  written  J. 

Three-tenths  is  written  yV 

One-third         "        J. 

Two-thirds      '' 

i 

One-fourth      ''        |. 

Three-fourths^' 

|. 

One-fifth          "        I. 

Four-fifths      '' 

f 

One-tenth        "        ^. 

Five-fourteenths     '' 

A. 

101.  Bead  the  following  : 

G^).      f      A      4       t 

J        f        1        ^ 

?. 

(b).       rV      AAA 

A     A     A     A 

A. 

(e).    a    H    a    H 

u    u   a   a 

il 

(^1).       U      If      «      If 

n   fi    «   Av 

AV. 

1.  What  is  J  of  6  ?    Of  10 

?     Of  16?     Of  18? 

2.  How  do  you  find  J  of  a  number  ? 

3.  What  is  i  of  6  ?     Of  9  ? 

Of  18?     Of  24? 

4.  How  do  you  find  ^  of  a  number  ? 

6.  What  is  J  of  8  ?     Of  12 

?     Of  16?     Of  20? 

6.  How  is  J  of  a  number  found  ? 

7.  What  is  ^  of  20  ?     Of  25  ?     Of  35  ?     Of  60  ? 

8.  What  is  ^  of  12  ?     Of  18  ?     Of  30  ?     Of  .42  ? 

Of  54? 

9.  What  is  1  of  21  ?     Of  42  ?     Of  35  ?     Of  56  ? 

Of  63? 

10.  What  is  i  of  4«  ?     Of  8a  ?    Of  10^  ?     Of  16b  ? 

11.   What  is  1  of  6c  ?     Of  ISb  ?     Of  21a  ?     Of  36a; 

? 

12.  What  is  ^  of  24  ?    Of  32:?;  ?    Of  48  ?   Of  56a  ? 

Of  72? 

13.  Are  there  as  many  ones  in  ^  of  24  as  there 

are  8's 

in  24  ? 

fJQ  SCHOOL  ARITHMETIC. 

14.  Since  ^  of  24  is  3,  how  many  are  3  eighths  of  24  ? 

15.  Find  f  of  40.     Of  32.     Of  16.    ^Of  48.     Of  64.    Of24^>. 

16.  Find  |  of  24.     Of  40.     J  of  32.*    Of  56.     Of  80. 

17.  What  is  ^  of  27  ?    |  of  27  ?    -J  of  45  ?    f  of  54  ? 

To  find  one  of  the  equal  parts  of  a  number  requires  division.  Thus, 
to  find  i  of  8  we  must  divide  8  by  2.  To  find  i  of  12  we  must  divide  12 
by  3 ;  that  is,  find  one  of  the  3  equal  parts  of  12. 

102.  The  first  13  of  the  following  examples  involve  the 
process  mentioned  in  (B).  The  quotient  in  each  is  concrete; 
it  corresponds  to  the  multiplicand. 

1.  Mr.  A  divided  $10  equally  between  two  boys.  How 
much  did  each  boy  get  ? 

Since  two  boys  get  $10,  each  boy  got  \  of  $10,  or  $5.     $10  -r-  2  =  $5. 

2.  I  paid  $1 5  for  3  chairs.     How  much  was  that  apiece  ? 

3.  If  4  bags  contain  28  bushels,  how  many  bushels  are  in 
each  bag  ? 

4.  John  earned  $30  in  6  weeks.  How  much  was  that  per 
week  ? 

5.  If  45  people  live  in  9  houses,  what  is  the  average  num- 
ber in  a  house  ? 

6.  How  much  was  calico  a  yard  when  8  yards  cost  88  cents  ? 

7.  A  merchant  paid  $96  for  8  stoves.  How  much  did  he 
pay  for  each  stove  ? 

8.  A  drover  has  84  cattle  in  7  stables.  How  many  has 
he  in  each  stablo  ? 

9.  There  are  10  rows  of  trees  in  an  orchard  containing  120 
trees.     How  many  trees  in  a  row  ?     In  2  rows  ? 

10.  For  9  clocks  a  jeweler  paid  $108.  How  much  did  he 
pay  for  one  clock  ?    For  2  ?     For  3  ? 

11.  If  5  hats  cost  $15,  how  much  will  8  hats  cost  ? 

12.  How  much  will  10  sheep  cost,  if  7  sheep  cost  $28  ? 

13.  A  lady  paid  25«  dollars  for  5  yards  of  fine  cloth.  How 
much  did  she  pay  for  one  yard  ? 


DIVISION.  71 

14.  When  wheat  is  h  cents  a  bushel,  liow  much  must  bo 
paid  for  a  busliels  ? 

15.  I  divided  GZ>  dollars  equally  among  3  boys,  and  then 
gave  one  of  the  boys  a  dollars  more.  How  much  had  that  boy 
then? 

103.  In  the  following  exercises  let  the  pupil  determine 
which  use  of  division  eacii  problem  illustrates ;  that  is, 
whether  one  of  the  equal  parts  or  the  number  of  them  is  to  be 
found. 

1.  At  $10  each,  how  many  carts  can  be  bought  for  $80  ? 

2.  If  8  coats  cost  $80,  how  much  does  one  cost  ? 

3.  My  milk  bill  for  one  week  was  9G  cents.  If  I  paid  8 
cents  a  quart,  how  many  quarts  did  I  get  ? 

4.  A  lady  gave  48  apples  to  13  boys.  How  many  apples 
did  each  boy  get  ? 

6.  If  8  apples  cost  24  cents,  what  is  the  cost  of  a.  dozen 
apples  ? 

6.  Seven  hunters  shot  84  rabbits.  What  was  each  cue's 
share  of  the  game  ? 

7.  If  5  tons  of  coal  cost  $30,  how  many  tons  can  be  bought 
for  $54  ? 

8.  Mr.  A  paid  110  cents  for  cloth  at  11  cents  a  yard,  and 
Mr.  B  paid  9G  cents  for  cloth  at  12  cents  a  yard.  How  many 
yards  did  both  buy  ? 

9.  One  jeweler  paid  $'«'2  for  watches,  and  another  paid 
$132  for  some  of  the  same  kind.  If  the  first  got  6  watches, 
how  many  did  the  second  get  ? 

10.  How  many  pencils  at  3  cents  each  can  be  purchased 
for  a  quarter  ?     How  many  cents  will  be  left  ? 

11.  At  4  cents  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  half  a  dollar  ? 

12  yards  cost  48  cents.  What  part  of  a  yard  will  the  2  cents  that 
remain  buy  ? 

12.  Why  is  not  8J  pencils  the  answer  to  the  10th  problem  ? 


Y2  SCHOOL  ARITHMETIC. 

13.  In  how  many  days  can  a  man  earn  $50,  if  he  receive 
$6  a  day  ? 

14.  How  many  sheep,  at  $7  a  head,  can  be  purchased  by  a 
man  who  lacks  $5  of  having  150  ? 

15.  What  is  the  cost  of  one  table,  if  4  tables  cost  $34  ? 

16.  In  how  many  days  can  a  man  earn  54a  dollars,  if  he 
receives  6«  dollars  a  day  ? 

17.  In  8  weeks  a  boy  earned  12h  dollars.     How  much  did 
he  earn  in  a  week  ?     In  2  weeks  ? 

WRITTEN     EXERCISES. 


104.  1.  Divide  8255 

byG. 

(a) 
6)8255 
13751 

(b) 
6)8255(1375  quotient 
6 

22 

18 
45 

42 
35 

30 
5 

For  convenience  we  write  the  divisor  at  the  left  of  the  dividend,  with 
a  line  between  them,  and  the  quotient  either  as  in  (a)  or  in  (b). 

In  8  there  is  one  6,  with  a  remainder  of  2.  Since  the  8  is  thousands, 
the  quotient  1  and  the  remainder  2  are  thousands.  2  thousands  =  20 
hundreds,  and  20  hundreds  and  2  hundreds  —  22  hundreds. 

In  22  there  are  three  6's,  with  a  remainder  of  4.  Since  the  22  is  hun- 
dreds, the  quotient  3  and  the  remainder  4  are  hundreds.  4  hundreds  = 
40  tens,  and  40  tens  +  5  tens  =  45  tens. 

In  45  there  are  seven  6's,  with  a  remainder  of  3.  Since  the  45  is  tens, 
the  quotient  7  and  the  remainder  3  are  tens.  8  tens  =  30  ones,  and  30 
ones  +  5  ones  =  35  ones. 

In  35  there  are  five  6's,  with  a  remainder  of  5,  which  may  be  written 
as  in  (a)  or  left  to  stand  as  in  (b). 
Pboof. — 1375  X  6  +  5  =  8255,  the  dividend,    Hence  the  work  js  correct. 


DIVISION. 

73 

The  process 

as 

sliown  in  (a)  is  < 

sailed  Short  Division;   as 

shown  in  (b)  it 

is 

called  Long  Division. 

Wherein  do  the  two 

processes  differ 

? 

Find  the  quotients  by  short  division  : 

2.  1237  -T-  3. 

18.  4405  -^  5. 

34. 

268735  H-  3. 

3.  1934  ^  4. 

19.  3264  -h  6. 

35. 

281076  -^  6. 

4.  2180  ^  5. 

20.  1896  -T-  8. 

36. 

340125  H-  5. 

6.  3265  -^  4. 

21.  6188 -=-4. 

37. 

532024  -^  4. 

6.  5375  -^  5. 

22.  1809  ^  9. 

38. 

659134  -^  7. 

7.  6894  -^  6. 

23.  2808  ^  9. 

39. 

386937  -^  9. 

8.  4506  -=-  3. 

24.  3629  H-  7. 

40. 

726859  -^  8. 

9.  6025  -f-  5. 

25.  8736  -f-  6. 

41. 

400002  ^  6. 

10.  6132  -^  4. 

26.  6320  ^  5. 

42. 

590023  -^  7. 

11.  4306  -V-  7. 

27.  9144-^8. 

43. 

646398  -^  9. 

12.  4955  ^  6. 

28.  5706  -^  6. 

44. 

200000  -f-  3. 

13.  5005  -^  5. 

29.  3800  -H  5. 

45. 

486018 --6. 

14.  1216  -^  8. 

30.  4008  -7-  8. 

46. 

$598524  - 

-$6. 

15.  2032  -^  4. 

31.  7398  --  9. 

47. 

$567014  - 

-$7. 

16.  7294  --  7. 

32.  4543  -^  7. 

48. 

$630927  - 

-$9. 

17.  3735  -^  3. 

33.  8888  -T-  6. 

49. 

1803449  - 

-$8. 

60.  9306^ 

-11 

53. 

10186  ^  11. 

51.  3432 -f 

-12. 

54. 

10188  -^  12. 

52.  5964 -f 

-12 

55. 

*69 

576  -^  $12. 

RULE     FOR     LONG     DIVISION. 

105.  Find  hotu  many  times  the  divisor  is  contained  in  the 
number  represented  by  the  fewest  left-hand  figures  of  the 
dividend  that  ivill  contain  it. 

Multiply  the  divisor  by  the  quotient  thus  obtained,  write 
the  product  under  the  left-hand  figures  used,  and  subtract. 

To  the  remainder  annex  the  next  figure  of  the  dividend,  and 
then  proceed  as  before,  until  all  the  figures  have  been  annexed. 

If  any  partial  dividend  is  less  than  the  divisor,  place  a 
cipher  in  the  quotient,  annex  the  next  figure  of  the  dividend, 
and  proceed  as  before. 


u 


SCHOOL  ARITHMETIC. 


Proof. — Multiply  the  divisor  by  the  quotient,  and  add  the  remainder, 
if  any,  to  the  product.  If  the  result  is  equal  to  the  dividend  the  work  is 
correct. 

Queries. — 1.  Why  must  each  remainder  be  less  than  the  divisor  ? 

2.  When  the  product  of  the  divisor  and  the  quotient  figure  is  greater 
than  the  partial  dividend  from  which  it  is  to  be  subtracted,  what  must  be 
done  ? 

1.  Divide  1728  by  24. 


24)1728(72 
168 
48 
48 

ones.     48  ones 
remainder  ? 


As  24  is  greater  than  17,  it  is  necessary  to  take  the 
number  represented  by  three  figures  of  the  dividend  for 
the  first  partial  dividend.  172  tens  -s-  24  =  7  tens,  and  a 
remainder.  24  x  7  tens  =  168  tens.  The  remainder  is  4 
tens,  and  the  new  dividend  is  4  tens  and  8  ones,  or  48 
^  24  =  2  ones.     24  x  2  ones  =  48  ones.      Is  there  any 


Find  the  quotients 

2.  4536  -T-  21. 

3.  9175  ^  25. 

4.  7998  -^  31.   • 

5.  7052  -^  41. 

6.  21879  -^-  51. 

7.  11792 -V- 22. 

8.  15136^32. 

9.  26250-^42. 

10.  42224  -r-  52. 

11.  37015 -^  55. 

12.  11914^23. 

13.  14058-^33. 

14.  14964 -^  43. 

41.  376859328  ^  48. 

42.  384710564  -i-  58. 

43.  238311937 -^  29. 

44.  287135862  h-  39. 

45.  429436049  ^  49. 

46.  '}f28395002  -^  59. 


by  long  division 

15.  13409-^53. 

16.  17400 -^  24. 

17.  21556  -V-  34. 

18.  22836  -^  44. 

19.  50058^54. 

20.  30660-^35. 

21.  32715-^45. 

22.  61620 -^  26. 

23.  22212-^36. 

24.  24472-^46. 

25.  32088-^56. 

26.  88020 -^  27. 

27.  16206  -^  37. 


28.  35344  h-  47. 

29.  21888 -^  57. 

30.  10388-^28. 

31.  18696  -^  38. 

32.  52542-^63. 

33.  10725-^75. 

34.  26487^81. 

35.  33943-^91. 

36.  43368  -=-  74. 

37.  66864-^84. 

38.  52776  ^  72. 

39.  25568  ~  68. 

40.  64108^94. 


47.  169135679 

48.  207407256 

49.  181481349 

50.  219752926 

51.  057662754 

52.  197393584 


137. 
168. 
147. 

178. 
201. 
304. 


DIVISION.  75 

63.  3049G3cS22  -^  400.  64.  108825952  ^  4528. 

54.  304123450  -f-  510.  65.  230884080  -^  5684. 

55.  442680498  -^  611.  66.  597126784  -^  6788. 

56.  584093472  ^  712.  67.  941108532  -^  7638. 

57.  263090640  -^  819.  68.  265283625  ^  8725. 

58.  383748326  -^  923.  69.  397046588  ^  9337. 

59.  677510968  -4-  937.  70.  308196056  -^  3962. 

60.  018875970  -f-  745.  71.  535673956  -^  8009. 

61.  353628594  ~  1023.  72.  810891081  -^  9009. 

62.  512763462  ^  2186.  73.  103031370  ~  8346. 

63.  498933150  --  3275.  74.  462017992  -^  4678. 

106.  1.  A  drover  bought  some  cattle  for  $17616.  If  the 
average  price  of  each  was  $48,  how  many  did  lie  buy  ? 

2.  The  salary  of  a  Congressman  is  $5000  a  year.  How 
much  is  that  a  day  ? 

3.  A  owes  B  $5200.  If  he  pays  him  $650  a  year,  in  how 
many  years  will  the  debt  be  canceled  ? 

4.  The  circumference  of  the  earth  is  about  8  million  rods, 
and  tliere  are  320  rods  in  a  mile.  How  many  miles  is  it 
around  the  earth  ? 

6.  A  grocer  bought  368  barrels  of  flour  for  $2208,  and  sold 
them  for  $2944.     How  much  did  he  gain  per  barrel  ? 

6.  TheW.  Y.  R.  R.  is  268  miles  long,  and  cost  $5,660,728. 
What  was  the  average  cost  per  mile  ? 

7.  The  salary  of  the  President  of  the  United  States  is 
$50000  a  year.     How  much  is  that  per  day  in  leap  year  ? 

8.  The  product  of  two  numbers  is  1,259,375.  One  of 
the  numbers  is  97  less  than  500.     What  is  the  other  ? 

9.  If  a  man  receives  $1600  a  year  for  his  labor,  and  spends 
$832,  in  how  many  years  can  he  save  enough  to  buy  a  farm 
of  132  acres,  at  $24  an  acre  ? 

10.  By  selling  a  farm  of  240  acres  for  $12720,  I  gained 
$1200.     How  much  did  I  pay  per  acre  for  the  farm  ? 

11.  How  often  can  436  be  subtracted  from  34444  ? 


76  SCHOOL  ARITHMETIC. 

12.  The  dividend  is  9689,  the  quotient  134,  and  the  re- 
mainder 41.     What  is  the  divisor  ? 

13.  A  double-track  street  railway  is  5  miles  long.  How 
many  rails  does  it  contain,  if  each  rail  is  24  feet  long,  there 
being  5280  feet  in  a  mile  ? 

14.  A  train  of  fifteen  cars  contained  279300  pounds  of 
flour  in  barrels.  How  many  barrels  were  in  each  car,  a  barrel 
of  flour  weighing  196  pounds  ? 

15.  Two  men  leave  Memphis,  Tenn.,  to  travel  around  the 
earth,  one  going  east  at  the  rate  of  154  miles  a  day,  the  other 
going  west  at  the  rate  of  144  miles  a  day.  In  how  many 
days  will  they  meet,  if  the  distance  around  is  16390  miles  ? 

16.  A  railroad  train  makes  2  round  trips  daily  between 
New  York  and  Philadelphia.  How  far  apart  are  these  cities 
if  the  train  runs  131400  miles  in  a  common  year  ? 

To  divide  by  lO,   lOO,   1000,  etc. 

107.  1.  80  H-  10  =  what  ?  750  ^  10  =  what  ?  3200  -^ 
10  =  what  ?  How  do  the  quotients  compare  with  the  divi- 
dends ? 

2.  700  -^  100  =  what  ?  2400  -^  100  =  what  ?  Compare 
quotients  with  dividends,  and  tell  how  the  latter  have  been 
changed. 

3.  75  -^  10  =  what  ?     What  is  the  remainder  ? 

4.  3825  -^  100  =  what  ?    -What  is  the  remainder  ? 

5.  43875  -i-  1000  =  what  ?     What  is  the  remainder  ? 
Where  are  these  remainders  seen  in  the  dividends  ?     Can 

the  quotients  be  seen  in  the  dividends  ?     Where  ? 

108.  Pkinciple. — A  number  may  be  divided  by  10,  100, 
1000,  etc.,  by  cutting  off  from  the  right  of  the  dividend  as 
many  figures  as  there  are  ciphers  in  the  divisor. 

The  part  cut  off  is  the  remainder,  and  the  rest  of  the  dividend  is  the 
quotient.  Thus,  4100  -^  100  =  41  ;  4125  -=-  100  =  41,  with  a  remainder 
25.     The  quotient  may  be  written  41fo\- 


DIVISION. 


YY 


Divide  the  following  : 


1.  380  by  10. 

2.  275  by  10. 

3.  420  by  10. 

4.  600  by  10. 
6.  775  by  10. 
6.  905  by  10. 


7. 

8. 

9. 
10. 
11. 
12. 


500  by  100. 

320  by  100. 

875  by  100. 
7200  by  100. 
2450  by  100. 
4315  by  100. 


13. 
14. 
15. 
16. 
17. 
18. 


6G000  by  1000. 

9300  by  1000. 

2460  by  1000. 

8725  by  1000. 
35000  by  1000. 
40009  by  1000. 


19.  Divide  31275  by  500. 


5 1 00)312  I  75 


Cutting  off  two  figures  from  the  right  of  the 
dividend  divides  it  by  100,  the  quotient  being 
312,  with  the  remainder  75.  Since  500  is  5 
timts  100,  the  quotient  is  5  times  as  large  as  it  should  be.  Hence  we 
divide  it  by  5,  getting  a  quotient  of  62  and  a  remainder  of  3,  which  is 
hundreds.  2  hundreds  4-  75  (the  first  remainder)  =  275,  the  entire  re- 
mainder.    Hence  the  quotient  is  62§^XB. 


Find  the  quotients  : 

26.  78960 

27.  62845 


80. 
90. 

28.  59320^300. 

29.  32856 -J- 500. 

30.  47623  -v-  600. 

31.  89974-^800. 


32.  102030  -^  900. 

33.  510075  H-  700. 

34.  246783  -^  800. 

35.  987654  -^  600. 

36.  100000-4-800. 

37.  808080  -h  700. 


20.  2765  -^  20. 

21.  4275  -r  30. 

22.  5180  -^  40. 

23.  3625  -4-  50. 

24.  7338  -^  60. 

25.  6774  -4-  70. 

38.  Divide  1728  by  12,  by  dividing  first  by  2,  and  then  the 
quotient  by  6. 

1728  4-  2  =  864.  864  4-  6  =  144. 

39.  Divide  2625  by  the  factors  of  15. 

40.  Divide  4536  by  21 ;  also  by  3  and  7,  and  compare  re- 
sults. 

109.  In  expressions  like  18  +  24  4-  6,  the  operation  indi- 
cated by  the  sign  -^  must  be  first  performed. 

Thus,  18  +  24-5-6  means  18  +  4,  not  42  -^  6.     120  -  80  -h  2  means 
120-40. 


78 


SOHOOL  ARITHMETIC. 


Find  the  value  of 

1.  43  +  18  ^  3. 

2.  19  -  36  -^  4. 

3.  27  ^  9  -1-  8. 

4.  35  -  21  -^  7. 


5.  22  +  9  X  3  -  49  -T-  7. 

6.  68  -  35  -^  5  +  7  X  10. 

7.  54  -  57  -^  19  -  3  X  17. 

8.  36  4-  7  X  9  -  63  -f-  9. 


GENERAL   PRINCIPLES   OF    DIVISION. 

no.  The  value  of  the  quotient  depends  upon  the  relative 
values  of  the  dividend  and  divisor.  Hence,  if  either  divi- 
dend or  divisor  is  changed,  the  quotient  will  be  changed. 
If  both  are  changed  equally  (as  to  ratio),  the  quotient  will 
not  be  changed,  as  may  be  seen  in  equations  (e)  and  (f) 
below. 

The  following  equations  illustrate  all  the  changes  : 

Given  Equation,  24  -^  6  =  4. 

(a)  Multiplying  the  divi- 
dend by  2  multiplies  the 
quotient  by  2. 

(b)  Dividing  the  dividend 
by  2  divides  the  quotient 
by  2. 

(c)  Multiplying  the  divisor 
by  2  divides  the  quotient 
by  2. 

(d)  Dividing  the  divisor  by 
2  multiplies  the  quotient 
by  2. 

(e)  Multiplying  both  divi- 
dend and  divisor  by  2  does 
not  change  the  quotient. 

(f)  Dividing  both  dividend 
and  divisor  by  2  does  not 
change  the  quotient. 


Changing 

(a)  48  ^    6  =  8. 

dividend. 

(b)  12  ^    6=2. 

Changing 

(c)  24  -^  12  =  2. 

divisor. 

(d)  24--    3  =  8. 

Changing 

(e)  48  -^  12  =  4. 

both 

equally. 

(f)  12  -T-    3  =  4. 

DIVISION.  79 

From  these  examples  are  deduced  the  following  general 
111.  Principles. — 1.  Multiplying  the  dividend  multiplies 
the  quotient,  and  dividing  the  dividend  divides  the  quotient, 

2.  Multiplying  the  divisor  divides  the  quotient,  and  divid- 
ing the  divisor  multiplies  the  quotient. 

3.  Multiplying  or  dividing  both  dividend  and  divisor  hy 
the  same  number  does  7iot  change  the  quotient. 

Queries. — 1.  If  a  number  equal  to  the  divisor  should  be  added  to  the 
dividend,  what  change  would  occur  in  the  quotient  ? 

2.  Subtracting  twice  the  divisor  from  the  dividend  would  have  what 
effect  on  the  quotient  ? 

3.  Would  adding  the  same  number  to  both  dividend  and  divisor  in- 
crease or  diminish  the  quotient? 

4.  If  the  same  number  were  subtracted  from  dividend  and  divisor, 
would  the  quotient  be  increased  or  diminished  ? 

5.  Does  subtracting  any  number  from  the  divisor  increase  or  diminish 
the  quotient  ? 


REVIEW   WORK. 


ORAL   EXERCISES. 


112,  1.  Whafc  number  is  represented  by  45  ? 

2.  What  number  is  represented  by  8  ones  of  the  first  period 
and  7  tens  of  the  second  period  ? 

3.  In  42  tens  liow  many  ones  ? 

4.  How  many  ones  in  3  hundreds,  6  tens,  and  5  ones  ? 

5.  If  9  hats  cost  127,  what  will  5  hats  cost  ? 

Since  9  hats  cost  $27,  1  hat  costs  -^  of  $27,  or  $3;  since  1  hat  costs 
$3,  5  hats  will  cost  5  times  $3,  or  $15. 

Query. — Why  will  5  hats  cost  5  times  $3  ? 

6.  If  6  sheep  cost  $30,  how  much  will  11  sheep  cost  ? 

7.  A  man  bought  10  books  for  $40,  and  sold  7  of  them  at 
the  same  rate.     How  much  did  he  receive  for  them  ? 

8.  How  much  will  12  yards  of  cloth  cost,  if  5  yards  cost 
55  cents  ? 

9.  How  much  will  13  pounds  of  meat  cost,  if  9  pounds 
cost  72  ceuts  ? 

10.  A  sold  5  pigs  and  B  sold  3,  each  getting  the  same 
price  per  head.     How  much  did  each  get  if  both  got  $64  ? 

11.  I  sold  a  calf  for  $19,  which  was  $7  more  than  it  cost 
me.     How  much  did  I  pay  for  it  ? 

12.  A  man  bought  6  barrels  of  flour  for  $30,  and  gave 
half  of  them  for  potatoes  at  $3  a  barrel.  How  many  barrels 
of  potatoes  did  he  get  ? 

13.  Which  is  cheaper,  and  how  much  per  dozen — eggs  at 
25  cents  a  dozen,  or  at  3  cents  each  ? 

14.  How  long  will  it  take  A  to  earn  $99,  if  he  earns  $18 
in  2  weeks  ? 


REVIEW   WORK.  81 

15.  How  many  days  can  three  men  live  on  the  provisions 
that  5  men  require  for  9  days  ? 

16.  If  7  men  can  dig  a  ditcli  in  9  days,  how  long  would  it 
take  3  men  ? 

17.  If  a  load  of  hay  lasts  8  cows  a  week,  how  long  would  it 
last  14  cows  ? 

18.  Twelve  times  7  are  how  many  times  4  ? 

19.  If  3  apples  are  worth  1  lemon,  and  2  lemons  are  worth 
13  pears,  how  many  pears  are  worth  1(S  apples  ? 

20.  When  rice  is  6  cents  a  pound,  how  many  pounds 
should  I  receive  in  exchange  for  9  dozen  eggs  at  a  cent 
apiece  ? 

21.  In  85  days  how  many  weeks  ? 

22.  How  many  days  in  8  weeks  and  5  days  ? 

23.  Jack  bought  a  dollar's  wortii  of  apples  at  the  rate  of 
2  for  5  cents.     How  many  did  he  get  ? 

24.  May  bought  2  dozen  eggs  at  25  cents  a  dozen,  and  sold 
them  at  the  rate  of  3  for  a  dime.     How  much  did  she  gain  ? 

25.  Owen  bought  9  oranges  for  7  cents  each  and  11  lemons 
for  5  cents  each  ;  he  gave  in  exchange  9  pounds  of  butter  at 
15  cents  a  pound.     How  much  was  due  him  ? 

26.  I  gave  half  a  dozen  dozen  pencils  worth  5  cents  each 
for  6  knives.     What  was  each  knife  worth  ? 

27.  How  many  letters  are  required  to  write  $2.41  in  words  ? 

28.  At  $2.88  a  dozen,  what  is  the  value  of  10  hoes  ? 

29.  If  a  boys  earn  ba  cents  per  day,  how  much  do  b  boys 
earn  in  one  day  ? 

30.  If  q  cows  eat  a  tons  of  hay  in  a  month,  how  much  will 
p  cows  eat  ? 

31.  John  was  h  years  of  age  a  years  ago.  How  old  will  he 
be  in  «  +  J  years  ? 

32.  At  %a  each,  what  will  be  the  cost  of  c  rabbits  ? 

33.  A  has  %b,  B  has  %c  more  than  A,  and  C  has  as  much  as 
the  difference  between  A^s  and  B's  money.  How  much  have 
they  together  ? 

0 


SCHOOL  ARITHMETIC. 


WRITTEN     EXERCISES. 


113.  1.  Miss  B  teaches  9  months  in  the  year  at  a  salary  of 
$1350.     How  long  does  it  take  her  to  earn  1900  ? 

2.  A  and  B  bought  a  farm  of  80  acres  for  $7360.  If  A 
paid  $3128,  how  many  acres  did  he  pay  for  ? 

3.  The  President  of  the  U.  S.  receives  $50000  a  year.  If 
his  salary  were  increased  $5  a  year,  how  much  would  he 
receive  a  day  ? 

4.  How  long  a  string  will  it  take  to  reach  around  a  barn  42 
feet  long  and  36  feet  wide  ? 

5.  Mr.  A  bought  a  piano  for  $450,  paying  one-half  in  cash, 
and  the  remainder  at  the  rate  of  $15  a  month.  If  he  made 
the  purchase  January  1,  1899,  when  did  he  make  the  last 
payment  ? 

6.  Rome  was  founded  753  years  before  the  birth  of  Christ. 
How  long  was  that  before  Columbus  discovered  America  ? 

7.  There  are  369600  feet  in  70  miles.  How  many  feet  in 
5  miles  and  a  half  ? 

8.  What  number  besides  269  will  exactly  divide  36853  ? 

9.  In  100  years  the  population  of  the  U.  S.  increased  from 
3,929,214  to  62,622,250.  What  was  the  average  increase  per 
year  ? 

10.  If  I  spend  a  quarter  a  day  for  books,  a  dollar  a  day  for 
rent,  and  $35  a  month  for  groceries,  how  much  do  I  save  in 
a  leap  year  if  my  salary  is  $2000  ? 

11.  Find  the  sum  of  the  five  largest  numbers  that  can  be 
expressed  by  the  figures  9,  8,  0,  4,  and  2. 

12.  The  minuend  is  7019,  the  remainder  3107.  The  sub- 
trahend is  how  many  times  the  sum  of  3,  2,  and  7  ? 

13.  The  divisor  is  437,  the.  quotient  86,  and  the  remainder 
50.     What  is  the  dividend  ? 

14.  Two  men  had  $7583  divided  between  them.  The  dif- 
ference between  their  shares  was  $223.  How  much  did  each 
man  get  ? 


REVIEW  WORK.  83 

16.  How  many  times  can  461  be  subtracted  from  57820, 
and  what  is  tlie  remainder  ? 

16.  Marvin  read  from  chapter  LXXVII  to  chapter  XCIX. 
How  many  chapters  did  lie  read  ? 

17.  How  many  quarts  of  oats  will  two  horses  eat  in  30  days, 
if  each  horse  eats  4  quarts  3  times  a  day  ? 

18.  The  skull  has  8  bones,  the  face  14,  the  ear  3,  the  trunk 
53,  the  shoulders  4,  an  arm  3,  the  wrists  16,  the  hands  38, 
the  legs  8,  the  ankles  14,  and  the  feet  36.  Allowing  33 
teeth,  how  many  bones  are  in  the  body  ? 

19.  If  a  lot  of  hay  lasts  18  horses  27  months,  how  long 
would  it  last  27  horses  ? 

20.  The  product  is  5832  and  the  multiplier  is  324.  What 
is  the  multiplicand  ? 

21.  If  a  horse  travels  8  miles  an  hour  and  a  locomotive  40 
miles  an  hour,  how  much  sooner  can  a  man  go  120  miles  by 
traveling  on  the  cars  than  by  going  on  horseback  ? 

22.  Tlie  product  of  three  numbers  is  13824.  Two  of  the 
numbers  are  18  and  32.     What  is  the  third  ? 

23.  Find  a  number  to  which  if  369  be  added  the  sum  will 
be  1001  less  than  9090. 

24.  A  man  bought  68  horses  at  $84  each  ;  11  of  them  died. 
At  what  price  must  he  sell  the  others  to  gain  $444  ? 

26.  Mr.  H  bought  160  acres  of  land  at  $75  an  acre.  After 
spending  $1200  dollars  for  improvements,  he  sold  it  at  a  gain 
of  $2000.     At  what  price  per  acre  did  he  sell  ? 

26.  A  newsboy  bought  papers  at  3  cents  each,  and  sold 
them  at  5  cents  each,  thereby  gaining  90  cents.  How  many 
papers  did  he  sell  ? 

27.  A  mile  is  5280  feet.  How  many  steps  of  2  feet  each 
will  a  boy  take  in  walking  5  miles  ? 

28.  How  many  years  does  it  take  to  make  the  difference 
between  saving  $2  a  month  and  $6  a  month  amount  to  a  sav- 
ing of  $100  ? 


84  SCHOOL  ARITHMETIC. 

29.  I  traded  120  head  of  cattle  at  164  a  head  for  1 60  acres 
of  land.     What  price  per  acre  did  I  pay  ? 

30.  If  8  horse  shoes  weigh  16  pounds,  how  many  horses 
can  be  shod  with  shoes  that  weigh  152  pounds  ? 

31.  Horace  rode  31680  yards  on  his  bicycle,  the  wheel  of 
which  was  12  feet  in  circumference.  How  many  turns  did 
the  wheel  make  ? 

32.  What  number  multiplied  by  twice  37  will  produce 
2664  ? 

33.  A  wagon  weighing  1000  pounds  contains  6  barrels  of 
flour  and  7  of  pork,  and  is  drawn  by  two  horses.  A  barrel  of 
pork  weighs  200  pounds  and  a  barrel  of  flour  196  pounds. 
How  many  pounds  does  each  horse  draw  ? 

34.  The  distance  from  Pittsburg  to  Philadelphia  is  354 
miles.  If  a  railroad  conductor  makes  a  round  trip  every  two 
days,  how  many  miles  does  lie  travel  in  4  weeks  ? 

35.  At  $15  per  uniform,  how  many  companies  of  85  soldiers 
each  can  be  uniformed  for  $10200  ? 

36.  The  smaller  of  two  numbers  is  3782,  and  their  differ- 
ence is  1218.     What  is  tlie  larger  number  ? 

37.  Mr.  E  paid  $125  an  acre  for  80  acres  of  coal  land.  He 
sold  the  coal  for  $64000,  and  the  land  at  $75  an  acre.  How 
much  did  he  gain  ? 

38.  How  many  weeks  would  a  man  take  to  walk  1344  miles, 
if  he  walks  4  miles  an  hour,  7  hours  a  day,  and  6  days  a 
week  ? 

39.  The  sum  of  18  equal  numbers  is  96346  less  than  a 
million.     Find  one  of  the  numbers. 


40.  What  is  the  value  of  175  +  92  x  105  ? 

41.  If  a  train  runs  28  miles  an  hour,  in  how  many  hours 
can  it  run  to  a  place  420  miles  distant  and  return  ? 

42.  A  miller  owing  $500  gave  in  part  payment  250  bushels 
of  wheat  at  $1.50  a  bushel,  and  paid  the  remainder  with  flour 
at  $5  a  barrel.     How  many  barrels  were  required  ? 


43.  A  drover  bought  45  liorses  at  $85  each,  and  sold  them 
so  as  to  gain  $720.  How  much  a  head  did  he  receive  for 
them  ? 

44.  An  extension  table  is  12  feet  long  when  its  four  boards 
are  in,  and  7  feet  long  when  they  are  out.  How  wide  is  each 
board  ? 

45.  Henry  was  30  years  old  when  Edward  was  born. 
Edward  was  14  years  old  in  1876.  How  old  will  Henry  be 
in  1910  ? 

46.  If  one  hen  lays  180  eggs  in  a  year,  liow  many  dozen 
eggs  slionld  2  dozen  hens  lay  in  2  years  Y 

47.  From  New  York  to  Havana  is  1260  miles,  from  Havana 
to  Aspinwall  is  1046  miles,  from  there  to  Panama  is  60  miles, 
and  from  Panama  to  San  Francisco  is  3616  miles.  What  is 
the  distance  between  San  Francisco  and  Havana  ? 

48.  A  party  of  64,  eight  more  than  half  of  whom  were 
ladies,  took  a  boat  ride  at  an  expense  of  $3  each.  If  all 
expenses  were  paid  by  the  gentlemen,  how  much  did  each 
pay  ? 

49.  I  paid  $7500  for  two  lots,  one  of  them  costing  $1000 
more  than  the  other.     What  did  I  pay  for  each  ? 

50.  A  and  B  together  have  $500,  and  A  has  $100  more 
than  B.      How  much  has  each  ? 

51.  The  area  of  Texas  is  265780  square  miles,  and  that  of 
Pennsylvania  is  45215  square  miles.  Into  how  many  States 
of  the  size  of  Pennsylvania  could  Texas  be  divided,  and  how 
many  square  miles  would  be  left  over  ? 

52.  At  $4  a  ton,  what  is  the  value  of  a  carload  of  coal 
weighing  17920  pounds,  counting  2240  pounds  to  a  ton  ? 

53.  There  are  640  acres  in  a  square  mile.  How  many  acres 
in  Rhode  Island,  whose  area  is  1250  square  miles  ? 

54.  Each  front  wheel  of  a  carriage  is  10  feet  in  circum- 
ference, and  each  hind  wheel  12  feet.  The  front  wheels 
will  make  how  many  more  turns  than  the  hind  wheels  in 
going  5  miles,  there  being  5280  feet  in  a  mile  ? 


B6  SCHOOL  ARITHMETIC. 

55.  If  steel  rails  weigh  72  pounds  to  the  yard,  and  SOOO 
pounds  are  a  ton,  how  many  tons  of  rails  will  be  required  to 
lay  2  miles  of  railroad,  half  of  which  is  to  have  double  track  ? 

SUPPLEMENTARY    EXERCISES.    (FOR    ADVANCED    CLASSES.) 

1 14.  1.  Prove  that  if  we  multiply  by  4,  and  divide  the 
product  by  100,  we  obtain  the  result  of  dividing  by  25. 

2.  Find  the  value  of  720  +  964  x  8  -  154  x  0  x  6. 

3.  Prove  that  $17  x  11  =  $11  x  17. 

4.  By  what  number  must  we  divide  a  given  number  to 
obtain  the  same  result  as  when  the  given  number  is  multi- 
plied by  2  and  divided  by  70  ? 

5.  The  remainder  is  723.  What  is  the  minuend  if  it  is 
twice  as  great  as  the  subtrahend  ? 

6.  The  minuend  is  «  +  a,  and  the  remainder  is  equal  to 
the  subtrahend.     Find  the  remainder. 

7.  What  number  multiplied  by  100  and  divided  by  4  gives 
the  same  result  as  is  obtained  by  multiplying  LS  by  25  ? 

8.  The  sum  of  two  numbers  is  60,  and  their  difference  is 
24.     What  are  the  numbers  ? 

9.  The  sum  of  two  numbers  is  a,  and  their  difference  is  d. 
What  are  the  numbers  ? 

10.  What  is  the  qnotient  when  $12  is  divided  by  $4? 
When  $«  is  divided  by  U  ? 

11.  A  man  living  at  the  rate  of  $3500  a  year  for  6  years 
finds  that  he  is  exceeding  his  income,  and  reduces  his  ex- 
penditures to  $2500  a  year.  At  the  end  of  4  years  he  finds 
that  he  is  just  out  of  debt.     What  is  his  income  ? 


FACTORS  AND  MULTIPLES. 

115.  1.  What  two  numbers  multiplied  together  will  make 
6  ?     Then  what  are  the  factors  of  G  ? 

2.  Is  each  factor  of  6  an  exact  divisor  of  G  ? 

116.  The  integers  which  multiplied  together  will  produce 
a  number  are  called  the  Factors  of  that  number. 

Thus,  5  and  6,  or  2,  3,  and  5  are  the  factors  of  30.  The  factors  of  a 
number  are  exact  divisors  of  it, 

1.  What  are  the  exact  divisors  of  8  ?  Of  7  ?  Of  10  ?  Of 
13  ?     Of  18  ?     Of  19  ?     Of  23  ? 

117.  A  number  whose  only  exact  divisors  are  itself  and  1 
is  called  a  Prime  Number. 

A  number  that  has  other  exact  divisors  is  called  a 
Composite  Nimiber. 

Thus,  3,  5,  11,  17,  etc.,  are  prime  numbers,  and  4,  9,  12,  20,  etc.,  are 
composite  numbers. 

1.  Make  a  list  of  all  the  prime  numbers  between  0  and  100. 

2.  Make  a  list  of  all  the  numbers  from  1  to  100  that  are 
exactly  divisible  by  2. 

3.  Make  a  list  of  all  the  numbers  between  0  and  144  that 
are  not  exactly  divisible  by  2. 

118.  A  number  that  is  exactly  divisible  by  2  is  called 
an  Even  Number.  All  other  numbers  are  called  Odd 
Numbers. 

119.  The  exact  divisors,  or  factors,  of  a  number  must  be 
found  by  inspection  or  by  trial.  The  following  facts  are 
very  helpful  in  finding  factors  : 


88  SCHOOL  ARITHMETIC* 

Any  number  is  exactly  divisible 

1.  By  2,  when  the  right-hand  figure  is  0,  2,  4,  6,  or  S. 

2.  By  d>  when  the  sum  of  the  numbers  represented  by  its 
digits  is  divisible  by  3. 

^  3.  By  4^  when  the  number  represented  by  the  two  right- 
hand  digits  is  divisible  by  4. 

4.  By  5^  when  the  right-hand  figure  is  0  or  5. 

5.  By  6,  when  it  is  divisible  by  2  and  3. 

6.  By  8,  when  the  number  represented  by  the  three  right- 
hand  digits  is  divisible  by  8. 

7.  By  9,  when  the  sum  of  the  numbers  represented  by  its 
digits  is  divisible  by  9. 

Find  some  divisors  of  the  following  by  inspection  : 


1. 

324. 

6.  8406. 

11. 

9072. 

16. 

84,306. 

2. 

]  75. 

7.  7300. 

12. 

8100. 

17. 

52,146. 

3. 

2G0. 

8.  2904. 

13. 

3285. 

18. 

93,528. 

4. 

513. 

9.  5344. 

14. 

7824. 

19. 

60,000. 

6. 

4Q0. 

10.  4563. 

15. 

5259. 

20. 

78,327. 

FACTORING. 

120.  1.  What  prime   numbers   multiplied   together    will 
produce  6  ?  10  ?  14  ?  22  ?  12  ? 

2.  What  prime  numbers  will  exactly  divide  18,  or  what  are 
the  prime  factors  of  18  ? 

121.  Prime  numbers  used  as  factors  are   called  Prime 
Factors. 

Thus,  3  and  7  are  the  prime  factors  of  21. 

1.  Can  11  and  18  both   be  divided  by  the  same  number  ? 
Can  12  and  25  ? 

2.  Which  of  the   numbers  in   the  preceding  example  are 
prime  numbers  ? 

122.  Two  numbers  that  have  no  common   factor  except 


FACTORINGS.  89 

tinity  are  said  to  be  Prime  to  each  other,  though  one  or 
both  of  them  may  be  composite. 

1.  Since  3  is  a  factor  of  G,  must  it  be  a  factor  of  two  6*s, 
or  12  ?     Of  three  6's,  or  18  ?     Of  any  number  of  G's  ? 

2.  Since  13  is  a  factor  of  36,  are  all  the  factors  of  12  also 
factors  of  36  ?     Find  by  trial. 

3.  Can  all  the  numbers  of  which  12  is  a  factor  be  exactly 
divided  by  the  factors  of  12  ?    Investigate. 

4.  Any  exact  divisor  of  a  factor  is  always  a  factor  of  what  ? 

123.  Principle. — A71  exact  divisor  of  a  factor  of  a  num- 
ber is  a  factor  of  the  mimber  itself. 

An  exact  divisor  may  be  a  fraction,  but  in  '*  factoring  ''  only  integral 
divisoi's  or  factors  are  considered. 

124.  The  process  of  finding  the  factors  of  a  number  is 
called  Factoring. 

WRITTEN    EXERCISES. 

125.  1.  What  are  the  prime  factors  of  360  ? 

2  360  Since  the  prime  number  2  is  a  divisor  of  360,  it  is 

o  TfiO  ^^^  ^^  ^^^  factors,  and  180  is  another.     Since  2  is  an 

exact  divisor  of  180,  it  is  a  factor  of  360  (Art.  123), 

2  90  as  is  90  also.     Since  2  is  an  exact  divisor  of  90,  it  is 

3  45  a  factor  of  360.      Likewise  3  and   5  being  exact 
„  ~rz  divisors  of    45  and  15    are    also    factors    of    360. 

Hence  2,  2,  2,  3,  3,  and  5  are  the  prime  factors  of 

5.  360. 

Note, — The  number  of  times  any  factor  occurs  in  a  product  may  be 
indicated  by  an  exponent.  Thus,  2^.  S'',  5  are  the  prime  factors  of  300. 
The  small  figures  written  above  and  to  the  right  of  the  factors  2  and  3 
are  exponents. 

Find  the  prime  factors  of  : 


2. 

60. 

8. 

480. 

14. 

2956. 

20. 

2310. 

3. 

108. 

9. 

672. 

15. 

4620. 

21. 

7644. 

4. 

144. 

10. 

1056. 

16. 

9170. 

22. 

64,384. 

5. 

180. 

11. 

1872. 

17. 

5432. 

23. 

20,000. 

6. 

315. 

12. 

2310. 

18. 

2002. 

24. 

242,424. 

7. 

308. 

13. 

3204. 

19. 

6006. 

25. 

714,510. 

LEAST     COMMON    MULTIPLE. 

126.  The  product  of  two  or  more  integers  is  called  a 
Multiple  of  those  numbers.  It  follows  that  any  number  is 
a  multiple  of  another  when  it  is  exactly  divisible  by  that 
number.     Every  number  is  a  multiple  of  its  factors. 

1.  Does  10  exactly  contain  both  2  and  5  ? 

2.  What  number  will  contain  7  and  3  without  a  re- 
mainder ? 

3.  Kame  a  multiple  that  is  common  to  5  and  11.  To  4 
and  6.     To  2,  3,  and  4. 

127.  A  multiple  that  is  common  to  two  or  more  numbers 
is  called  a  Common  Multiple. 

1.  What  is  the  least  number  that  will  exactly  contain  3 
and  5?    4  and  6  ?    2,  3,  and  4  ? 

2.  Is  24  a  common  multiple  of  3  and  4  ?  Is  it  their  smallest 
common  multiple  ?     What  is  their  least  common  multiple  ? 

128.  The  least  number  that  is  exactly  divisible  by  each 
of  two  or  more  numbers  is  called  their  Least  Common 
Multiple,  ivritten  L.  C.  M. 

1.  What  are  the  prime  factors  of  G  ?  Of  10  ?  What  is 
their  L.  CM.? 

2.  What  are  the  prime  factors  of  30  ?  How  do  they  com- 
pare with  those  of  6  and  10  ? 

3.  What  factor  is  common  to  6  and  10  ?  Does  it  occur 
twice  in  the  factors  of  30  ? 

4.  Since  30  contains  both  6  and  10,  must  it  contain  all 
their  prime  factors  ? 


LEAST  COMMON  MULTIPLE.  01 

5.  The  factors  of  0  are  2  and  3,  and  those  of  14  are  3  and 
7.  Which  of  these  factors  must  be  multiplied  together  to 
produce  the  L.  C.  M.  of  6  and  14  ? 

129.  Principle. — The  least  common  multiple  of  two  or 
more  numbers  contains  all  the  prime  factors  of  those  number s, 
and  no  others. 

If  a  factor  is  common  to  two  or  more  numbers,  it  is  contained  in  the 
L.  C.  M.  only  the  greatest  number  of  times  it  enters  into  any  one  of  the 
numbers — not  as  often  as  it  occurs  in  all  of  them. 

WRITTEN     EXERCISES. 

130.  1.  Find  the  least  common  multiple  of  25,  30,  and  42. 
25  =  5  X  5.  The  least  common  multiple  must 
30  =  2  X  3  X  5.  contain  all  the  prime  factors  of  25,  30, 
42  =  2  X  3  X  7.  ajMl  42,  that  is,  2,  3,  5,  and  7.  Each 
2x3x5x5x  7  =  1050.      of  these  must  be  contained  as  often  as 

it  occurs  in  any  one  set  of  factoi-s. 
The  only  factor  that  occurs  twice  in  one  number  is  5.     Hence  the 
factors  of  the  L.  C.  M.  are  2,  3,  5,  5,  7.     Their  product  is  1050,  the 
L.  C  M.     The  following  method,  which  is  in  common  use,  is  based  upon 
the  same  principle : 

Since  2  is  an  exact  divisor  of  some  of  the  num- 
bers, it  is  a  factor  of  the  L.  C.  M.  Since  3  is  an 
exact  divisor  of  some  of  the  quotients,  it  is  a 
factor  of  the  L.  C.  M.  (Art.  129).  We  find  in  the 
same  manner  that  5  is  also  a  factor  of  the  L.  C.  M. 
The  last  quotients  which  are  prime  to  each  other  are  also  factoi-s  of  the 
L.  C.  M.     Hence  the  L.  C.  M.  is  3  x  3  x  5  x  5  x  7,  or  1050. 

Find  the  L.  C.  M.  of  the  following  : 

2. 

3. 

4. 

5. 

6. 

1.  When  one  of  the  numbers  is  a  factor  of  another,  it  may  be  dis- 
regarded, as  its  multiple  contains  the  same  factors. 

2.  When  several  numbers  have  no  common  factor,  their  product  is 
the  L.  C.  M. 


2 

25, 

30, 

42. 

3 

25. 

15, 

21. 

5 

25, 

0, 

7. 

5, 

1, 

12, 

24, 

30. 

7.  4, 

5, 

9, 

8, 

12, 

6. 

18, 

97 

32. 

8.  7, 

0 

3, 

4, 

5, 

6. 

22, 

33, 

55. 

9.  6, 

7, 

8, 

10, 

14, 

16. 

28, 

30, 

60. 

10.  4, 

6, 

8, 

16, 

24, 

48. 

36, 

50, 

70. 

11.  3, 

5, 

7 

11, 

13, 

17. 

CANCELLATION. 

131.  Cancellation  is  a  process  of  shortening  the  work  in 
problems  that  involve  multiplication  and  division.  It  is 
based  on  two  principles. 

1.  What  is  the  product  of  6  x  5  ?  Of  3  x  5  ?  How  do 
the  products  compare  ? 

2.  What  is  the  product  of  8  x  2  ?  Of  4  x  2  ?  How  do 
the  products  compare  ?  How  could  you  get  the  second 
product  from  the  first  ? 

3.  Does  dividing  one  factor  by  any  number  divide  the 
product  by  the  same  number  ?     Find  by  trial. 

132.  Principles. — 1.  Dividing  any  factor  of  a  series 
of  factors  hy  any  number  divides  the  product  by  the  same 
number. 

2.  Dividing  both  dividend  and  divisor  by  the  same  number 
does  not  change  the  qnotient.     (Art.  110.) 

WRITTEN     EXERCISES. 

133.  1.  Divide  the  product  of  3,  21,,  and  25  by  tlie  prod- 
uct of  3,  7,  and  10. 

3         5  The  division  is  indicated  by  writing  the  dividend 

3  X  ^Z  X  ^^  above  and  the  divisor  below  the  line.  Dividing  both  by 
3  X  J  X  ^0  3  cancels  that  common  factor.  Dividing  both  by  7 
2  cancels  7  in  the  divisor  and  21  in  the  dividend,  leaving 
the  qnotient  3  in  the  latter.  Dividing  both  by  5 
cancels  10  and  25,  leaving  the  quotient  2  in  the  divisor  and  the  quotient 
5  in  the  dividend.  The  product  of  the  remaining  factors  of  the  dividend 
is  15.     Hence  the  quotient  is  15  -5-  2,  or  7|. 


CANCELLATION.  93 

Find  the  quotients  of  the  following  : 

^   4  X  5  X  6  X  10  45  X    8  X  11  X  73 


3. 


2  > 

:   3  X 

5  X  8  * 

8  X  14  X  9  X  12 

6  X  15  X  7  X  4 

12 

X  8 

X  9  X  30 

4 

X  80 

X  6  X  9  * 

25 

X  32 

X  18  X  7 

16 

X  15 

X  28  X  9 

13 

X  14 

X  15  X  IG 

24  X 

18  X 

15  X 

33' 

()3  X 

13  X 

93  X 

23 

39  X  ^1  X 

21  X 

69* 

121 

X  54 

X  28 

X  35 

44 

X  219 

X  30 

84  X 

65  X 

55  X 

49 

56  X 

63  X 

70  X 

22' 

132 

X  52  X  68  X  45 

10. 


6. .  11. 

7  X     8  X     9  X  10  77  X  65  X  51  X  20 

12.  How  often  is  12  x  13  x  50  contained  in  65  x  10  x 
84  X  3  ? 

13.  How  many  pounds  of  butter  at  28  cents  a  pound  must 
be  paid  for  25  yards  of  cloth  at  56  cents  a  yard  ? 

14.  How  many  barrels  of  apples,  each  containing  3  bushels, 
worth  70  cents  a  bushel,  are  worth  as  much  as  20  boxes  of 
crackers,  containing  15  pounds  each,  if  2  pounds  are  worth 
28  cents  ? 

15.  A  miller  sold  20  barrels  of  flour,  196  pounds  each,  at 
3  cents  a  pound,  and  received  his  pay  in  wheat  at  84  cents  a 
bushel.  If  there  were  2  bushels  in  a  bag,  how  many  bags 
did  he  get  ? 

16.  If  36  men,  working  8  hours  a  day,  can  do  a  piece  of 
work  in  57  days,  how  long  would  it  take  27  men,  working  9 
hours  a  day  ? 

17.  The  factors  of  the  dividend  are  10,  14,  9,  25,  and  32 ; 
the  factors  of  the  divisor  are  5,  16,  7,  and  25.  What  is  the 
quotient  ? 


UNITED    STATES    MONEY. 

(An  Introduction  to  Decimal  Fractions.) 

134.  United  States  money  has  a  decimal  currency.  It  is 
written  as  dollars  and  decimal  parts  of  a  dollar,  called  dimes, 
cents,  and  mills. 

1.  A  dime  is  what  part  of  a  dollar  ?  How  is  it  written  ? 
($.1.)  How  may  this  be  read?  (One  tenth  of  a  dollar.) 
Then  how  would  you  write  2  dimes,  or  3  tenths  of  a  dollar  ? 

2.  How  is  3  tenths  of  a  dollar  written  ?  5  tenths  ?  7 
tenths  ?     9  tenths  ?     8  dimes  ?     15  dimes  ?     25  tenths  ? 

3.  Write  3  dollars  and  5  dimes.  7  dollars  and  9  tenths  of 
a  dollar.  20  dollars  and  3  tenths  of  a  dollar.  5  and  one- 
tenth  dollars. 

4.  What  is  always  written  in  the  first  place  to  the  right  of 
dollars  ?  (Tenths  of  a  dollar.)  What  separates  the  dollars 
from  the  tenths  of  a  dollar  ? 

5.  How  is  $.01  read  ?  A  cent  is  what  part  of  a  dollar  ? 
Then  how  would  you  write  2  cents,  or  2  hundredths  of  a 
dollar  ? 

6.  Write  3  cents,  or  3  hundredths  of  a  dollar.  5  hun- 
dredths.    7  hundredths.     8  cants.     9  hundredths. 

7.  Write  2  dollars  and  6  cents.  5  dollars  and  8  hundredths 
of  a  dollar.  9  dollars  and  9  hundredths.  Why  put  a  cipher 
between  dollars  and  hundredths  ? 

8.  What  is  always  written  in  the  second  place  to  the  right 
of  dollars  ?     (Hundredths  of  a  dollar.) 

9.  Since  there  are  ten  cents  in  a  dime,  what  is  the  difference 
between  %.\  and  $.10  ?    Are  they  read  in  the  same  way  .^ 


UNITED  STATES  MONEY.  95 

10.  How  many  hundredths  of  a  dollar  in  a  tenth  of  a  dollar  ? 
In  2  tenths  ?     In  2  tenths  and  5  hundredths  ? 

11.  $.25  may  be  read  25  cents  ;  or  2  dimes  and  5  cents  ;  or 
2  tenths  and  5  hundredths,  or  25  hundredths  of  a  dollar. 

12.  A  cent  is  what  part  of  a  dime  ?  Then  a  huiidredtli  of 
a  dollar  is  what  part  of  a  tenth  of  a  dollar  ?  One  tenth  is 
equal  to  how  many  hundredths  ? 

13.  In  $.22,  which  2  has  the  greater  value  ?  Its  value  is 
how  many  times  the  value  of  the  other  ? 

14.  Write  24  hundredths  of  a  dollar,  3  tenths  and  5 
hundredths.  How  many  tenths  of  a  dollar  can  be  written 
with  one  figure  ?   How  many  hundredths  ?    With  two  figures  ? 

15.  How  is  S.OOl  read  ?  Since  there  are  1000  mills  in  a 
dollar,  what  part  of  a  dollar  is  1  mill  ?  Then  how  is  2  mills, 
or  2  thousafidths  of  a  dollar,  written  ? 

16.  Write  3  mills,  or  3  thousandths  of  a  dollar.  5  thou- 
sandths. 9  thousandths.  7  mills.  5  dollars  and  5  thou- 
sandths of  a  dollar. 

17.  What  is  always  written  in  the  third  place  to  the  right 
of  dollars  ?     (Thousandths  of  a  dollar.) 

18.  How  many  mills  in  a  cent  ?  Then  how  many  thou- 
sandths of  a  dollar  in  a  hundredth  of  a  dollar  ?  In  10  hun- 
dredths, or  a  tenth  ?  One  dollar  equals  how  many  tenths  of 
a  dollar?   How  many  hundredths  ?    How  many  thousandths  ? 

19.  $.375  maybe  read  375  mills;  or  37  cents,  5  mills  ;  or  3 
dimes,  7  cents,  5  mills  ;  or  3  tenths,  7  hundredths,  5  thou- 
sandths of  a  dollar,  or  375  thousandths  of  a  dollai'. 

20.  Since  5  mills  equal  half  a  cent,  $.375  may  be  read  37 
and  one-half  cents,  and  written  $.37 J. 

135.  Copy  and  complete  the  following  : 

1.  7  dimes,  or    7     tenths  of  a  dollar  =  $.7,  or  $.70. 

2.  3  dimes,  or ''         "       "      =  (  ). 

3.  5  cents,  or  "         ''       "      =  (  ). 

4.  9  cents,  or  —      ''        ''      **     =  (  ). 


96  SCHOOL  ARITHMETIC. 

136.  The  processes  of  adding  and  subtracting  in  U.  S. 
money  are  the  same  as  in  simple  numbers,  or  integers. 
Dolhirs  should  be  written  under  dollars,  cents  under  cents, 
mills  under  mills.  The  decimal  points  should  be  in  a  vertical 
line. 

Perform  the  operations  indicated  : 

1.  172.65  +  $18.23.  8.  $G  +  $.6  +  $.08 +  $.008. 

2.  $39.47  +  $26.82.       9.  $271.83  -  $187.93. 

3.  $53.90  +  $18.25.  10.  $1,000  -  $100.75. 

4.  $91.03  +  $4,775.  11.  $86.37  -  $24.80. 

5.  $.325  +  $10,584.  12.  $9,875  -  $5,312. 

6.  $.875  +  $.75  +  $.093.  13.  $5,003  -  $2,008. 

7.  $3.40  -f  $.205  +  $80.  14.  $100  -  $.975. 

15.  $73,806  +  $16,194  -  $89.98. 

16.  $1  +  $.1  -  $.01  +  $10. 

17.  To  3  tenths  of  a  dollar  add  7  hundredths  of  a  dollar. 

18.  From  7  tenths  of  a  dollar  subtract  35  hundredths  of  a 
dollar. 

19.  What  is  the  difference  between  4  hundredths  of  a 
dollar  and  9  tenths  of  a  dollar  ? 

20.  From  the  sum  of  8  tenths  and  5  thousandths  of  a 
dollar  subtract  9  hundredths  of  a  dollar. 

21.  A  farmer  bought  two  cows,  giving  $29.50  for  one  and 
$36.75  for  the  other.  He  gave  in  payment  a  wagon  worth 
$42.25,  and  the  rest  in  cash.  How  much  money  did  he 
give  ? 

22.  One  month  a  man  worked  24  days  at  $2.75  a  day.  His 
expenses  were  $41.70.     How  much  did  he  save  ? 

Dollars  and  decimal  parts  of  a  dollar  are  multiplied  and  divided  just 
as  integers  are.  Care  must  be  taken  to  point  off  the  proper  number 
of  places  for  cents  and  mills, 

23.  From  $12,375  x  25  subtract  12  times  $20.50. 

J37.  United  States  money  has  a  decimal  scale;  that  is, 


UNITED  STATES  MONEY.  97 

1  of  any  order  or  denomination  is  equal  to  10  of  the  next  lower 
order. 

Thus,  $1  =  10  dimes  ;  1  dime  =  10  cents  ;  1  cent  =  10  mills. 

1.  How  many  cents  in  5  dimes  ?     How  many  mills  ? 

2.  How  many  dimes  in  4  dollars  ?  How  many  cents  ? 
How  many  mills  ? 

138.  Carefully  examine  the  following  : 

$        dimes      cents      mills 

(a)  8.       =  80.     =  800.  =  8000. 

(b)  3.25  =  32.5  =  325.  =  3250. 

Query. — 1.  In  (a),  how  have  we  changed  dollars  to  lower  denomina- 
tions ?    By  annexing  what  ? 

2.  In  (b),  how  have  the  changes  been  made  ?  By  moving  what  ?  In 
which  direction  ? 

Principle. — Any  denomination  is  changed  to  a  lower  hy 
annexing  one  or  more  ciphers,  or  by  moving  the  decimal  point 
one  or  more  places  to  the  right. 

Change  to  mills,  or  thousandths  of  a  dollar  : 

1.  15.  6.  370  dimes.  11.  12.05. 

2.  3  dimes.  7.  435  cents.  12.  $6,005. 

3.  9  cents.  8.  $7.65.  13.  $.34. 

4.  21  dimes.  9.  $1.5.  14.  $2,125. 

5.  75  cents.  10.  7  dollars.  16.  $.875. 

139.  Carefully  examine  the  following  : 

mills        cents        dimes  $ 

(a)  2000    =200      =20        =2 

(b)  3125.  =  312.5  =  31.25  =  3.125. 

Query. — 1.  In  (a),  how  have  we  changed  mills  to  higher  denomina- 
tions ?    By  cutting  off  what  ? 

2.  In  (b),  how  have  the  changes  been  made  ?  By  moving  what  ?  In 
which  direction  ? 

Prikciple.i^^w^/  denomination  is  changed  to  a  higher  by 
cutting  off  one  or  more  ciphers,  or  by  moving  the  decimal  point 
one  or  more  places  to  the  left. 

7 


98  SCHOOL  ARITHMETIC. 

Change  to  dollars,  or  to  dollars  and  decimal  parts  of  a 
dollar  : 

1.  7000  mills.  7.  865  cents. 

2.  500  cents.  8.  750  mills. 

3.  80  dimes.  9.  1000  dimes. 

4.  600  dimes.  10.  8625  mills. 

5.  9000  cents.  11.  1250  cents. 

6.  35  dimes.  12.  12375  miHs. 

140.  It  was  learned  in  multiplication  of  integers  that  an- 
nexing one  cipher  to  a  number  multiplies  it  by  10  ;  annexing 
two  ciphers,  multiplies  it  by  100  ;  and  so  on.  The  same  is 
true  in  IT.  S.  money. 

Thus,  $2.00  X  10  =•  $20.00  ;  $2.00  x  100  =  $200.00. 

141.  Moving  the  decimal  point  one  place  to  the  right 
multiplies  expressions  of  U.  S.  money  by  10  ;  moving  it 
two  places,  multiplies  by  100  ;  and  so  on. 

Thus,  $3.25  X  10=  $32.50  ;  $3.25  x  100  =  $825. 

142.  Moving  the  decimal  point  one  place  to  the  left  divides 
expressions  of  U.  S.  money  by  10 ;  moving  it  two  places, 
divides  by  100  ;  and  so  on. 

Thus,  $125.50  -- 10  =  $12.55  ;  $125.50  -?- 100  =  $1,255. 
Find  the  value  of  the  following  : 

(Omitting  the  dollar  mark  does  not  affect  the  operation.) 

1.  $52  X  10.  8.  13.245  x  10.  15.  $300  x  100. 

2.  $7.25  X  10.  9.  $.625  x  100.  16.  $60.50  x  100. 

3.  $1.50  X  100.  10.  $.004  X  1000.  17.  $1,627  x  1000. 

4.  $.75  X  100.  11.  $330.00  -^  $10.  18.  $700.00  -^  $100. 

5.  $4.07  X  10.  12.  73.5  -f-  10.     '  19.  300.00  -^  100. 

6.  $6  X  100.  13.  47.3  -^  100.  20.  82.50  -4-  10. 

7.  8.25  X  100.  14.  219  -^  100.  21.  513.7  ~  100. 


FRACTIONS. 

143.  1.  In  measuring  milk  Kate  uses  a  can  that  holds  half 
a  gallon.  She  fills  it  twice.  How  many  lialf-gallons  has  she  ? 
How  many  gallons  ? 

2.  A  merchant  measures  a  piece  of  silk  with  a  ruler  one 
third  of  a  yard  long,  and  finds  the  piece  to  contain  two  thirds 
of  a  yard.    How  many  times  did  he  apply  the  measuring  unit  ? 

(a).  Is  the  piece  of  silk  a  yard  in  length  ? 
(b).  What  unit  of  measure  did  he  use  ? 
(c).  How  many  times  did  he  take  the  unit  ? 

3.  A  grocer  selling  molasses  filled  a  jar  three  times.  If  the 
jar  held  one  fourth  of  a  gallon,  how  much  molasses  did  he 
sell  ? 

(a).  What  measuring  unit  did  he  use  ? 

(b).  What  number  tells  how  many  times  he  took  or 
repeated  the  unit  ? 

(c).  Is  the  unit  of  measure  one  of  the  equal  parts  of  a 
gallon  ? 

(d).  What  expresses  the  quantity  of  molasses  sold  ? 

4.  A  farmer  used  a  half -bush  el  to  measure  his  wheat,  filling 
it  five  times. 

(a).  What  was  the  measuring  unit  ? 
(b).  How  many  such  units  in  a  bushel  ? 
(c).  How  many  half-Mishels  had  he  ? 

144.  A  Fraction  is  a  number  whose  unit  of  measure  is 
one  of  the  equal  parts  of  a  certain  whole  or  quantity. 

Thus,  three  fourths  of  a  yard  (3  fourth-yards)  is  a  fraction,  its  unit  of 
measure  being  one  fourth-yard — one  of  the  four  equal  parts  of  a  yard. 


100  SCHOOL  ARITHMETIC.         ^ 

Five  half-bushels  (5  halves  of  a  bushel)  is  a  fraction  ;  its  unit  of  measure 
is  one  half-bushel — one  of  the  two  equal  parts  of  a  bushel. 

1.  An  integer  (Art.  6)  is  a  number  whose  unit  of  measure  is  an  entire 
quantity — not  one  of  the  equal  parts  of  a  larger  quantity. 

2.  The  fraction  f  yard  may  be  regarded  as  3  fourth-yards  or  as  three 
fourths  of  a  yard.  The  unit  of  measure  is  one  of  the  four  equal  parts  of 
tiyard,  and  this  unit  is  repeated  3  times  in  measuring  the  quantity,  which 
compared  with  a  yard  is  three  fourths  as  great. 

145.  The  general  method  of  expressing  fractions  is  by 
two  numbers,  written  one  above  the  other,  with  a  line  be- 
tween them  ;  as,  |.  But  a  special  class  of  fractions,  called 
Decimal  Fractions,  is  expressed  in  a  notation  peculiar  to 
themselves. 


DECIMAL.    FRACTIONS. 

146.  1.  When  anything  is  divided  into  ten  equal  parts, 
what  is  one  part  called  ? 

2.  When  each  of  these  ten  parts  is  divided  into  ten  equal 
parts,  how  many  parts  are  there  ?  What  is  one  part 
called  ? 

3.  When  each  of  these  100  parts  is  divided  into  ten  equal 
parts,  how  many  parts  are  there  ? 

147.  AVhen  anything  is  divided  into  tenths,  limidredtJis, 
thousandths,  etc.,  the  parts  are  called  Decimal  Parts;  that 
is,  tenth-parts,  the  word  decimal  being  derived  from  decern, 
the  Latin  word  for  ten. 

148.  A  Decimal  Fraction  is  a  number  whose  unit  of 
measure  is  one  of  the  decimal  or  tenth  parts  of  a  certain 
quantity. 

Thus,  9  tenths  (unit  1  tenth),  25  hundredths  (unit  1  hundredth),  13 
thousandths  (unit  1  thousandth),  etc.,  are  decimal  fractions. 

Decimal  fractions  are  often  called  simply  decimals. 


5^. 


FRACTIONS.  ,"%H 


149.  The  decimal  fractions  one  tenths  one  hundredth,  one 
thousandth,  etc.,  are  obtained  by  dividing  a  quantity  or 
whole  into  10  equal  parts  {tenths),  and  each  of  these  into 
10  equal  parts  {hundredths),  and  each  of  these  again  into  10 
equal  parts  {thousandths) ;  hence, 

1  (whole)  =  10  tenths, 

1  tenth  =  10  hundredths, 

1  hundredth  =  10  thousandths. 

10  thousandths  =  1  hundredth, 
10  hundredths  =  1  tenth, 
10  tenths  ==  1  (whole). 

150.  It  is  seen  that  tenths,  hundredths,  thousandths, 
etc.,  taken  in  order,  decrease  in  value  from  left  to  right 
by  the  scale  of  tens,  just  as  integers  do.  That  is,  1  in  any 
place  or  order  is  equal  to  10  in  the  next  place  to  the 
right ;  and  10  in  any  place  is  equal  to  1  in  the  next  place  to 
the  left. 

151.  Since  the  notation  of  decimals  follows  the  same  law 
as  that  of  integers,  an  integer  and  a  decimal  fraction  may  be 
written  as  one  expression,  as  in  U.  S.  money. 

The  first  place  to  the  right  of  ones  is  tenths ;  the  second, 
hundredths;  the  third,  thousandths,  etc. 

152.  A  point  (.),  called  the  Decimal  Point,  is  placed 
before  tenths  to  locate  ones. 

Thus,  three  tenths  is  written  .3 

153.  An  integer  and  a  decimal  written  together  as  one 
number  is  called  a  Mixed  Number. 

In  writing  mixed  numbers  the  decimal  point  is  placed  between  the  in- 
tegral part  and  the  fractional  part,  thus  :  6.9:  2.03.  When  there  is 
no  integral  part,  what  may  be  written  in  ones'  place  ? 


■jte:; 


SCHOOL   ARITHMETIC. 


154.  The  relation  of  decimals  to  integers  is  clearly  shown 
by  the  following  diagram  : 


] 

NTEGERS 

Decimals 

00 

00 

fl 

f> 

.„ 

e8 

2       w 

-4-3 

f3 

1 

Etc. 
Millions 

£  i 

9          = 

3 

i  ^ 

1  i 

,      o 

00 

1 

Q 

M 

O 

o 

w 

to 

-t-3 

B 

Thousandths 

Ten-thousand 

Hundred-thoi 

Millionths 

Etc. 

( 

/) 

(e 

)(d 

)W 

1 

«f) 

J«) 

1 

WO 

i)(, 

0  (. 

n 

It  will  be  noticed  that : 

1.  Orders  at  equal  distances  to  the  left  and  right  of  ones^ 
place  have  corresponding  names ;  the  names  at  the  right 
having  the  fractional  ending  ths. 

2.  The  increase  and  decrease  according  to  the  decimal  scale 
go  right  along,  without  regard  to  the  decimal  point. 

3.  For  every  jy«^r^  of  the  unit  expressed  by  any  order  on 
the  right,  there  is  a  corresponding  multiple  of  the  unit  ex- 
pressed by  the  corresponding  order  on  the  left. 

4.  All  orders,  both  higher  and  lower,  are  derived  from 
ones.  Tens  denotes  tens  of  ones,  and  tenths  denotes  tenths 
of  one. 

READING    DECIMALS. 

156.  In  reading  integers,  we  give  the  numbers,  but  omit 
the  name ;  thus,  25  is  read  tweyity-five.  The  name  omitted 
is  ones,  which  is  the  name  of  the  right-hand  order  of  the 
integer. 


FRACTIONS.  103 

150.  In  reading  decimals,  we  give  both  number  and 
name.  Thus,  .25  is  read,  not  twenty-five^  but  twenty-five 
hundredths.  Tlie  name  given  is  hundredths,  which  is  the 
name  of  the  rifrht-hand  order  of  the  decimal. 


With  the  exception  of  giving  the  name,  decimals  are  read  pre- 
cisely as  integers  are  read. 

Rule. — Read  the  decimal  as  an  integral  number,  and  give 
it  the  name  of  the  right-hand  order. 

Read  the  following  decimals  : 


1. 

.3. 

6. 

.87. 

11. 

.23742. 

16. 

.003761. 

2. 

.9. 

7. 

.087. 

12. 

.00013. 

17. 

.009042. 

3. 

.15. 

8. 

.235. 

13. 

.00304. 

18. 

.0007103. 

4. 

.35. 

9. 

.101. 

14. 

.01238. 

19. 

.00000001. 

6. 

.07. 

10. 

.3005. 

15. 

.000013. 

20. 

.327604385, 

1.  In  reading  mixed  numbers,  read  the  integral  part  first,  then  the 
decimal  part,  connecting  them  with  the  word  and.  Thus,  205.03  is 
read,  two  hundred  five  and  three  hundredths. 

2.  It  is  sometimes  necessary  to  make  a  pause  before  giving  the  name 

of  the  decimal.    Thus,  in  .300,  read  three  hundred thousandths  ;  and 

in  .00003,  read  three hundred-thousandths. 

3.  Expressions  like  .12^  and  .33^^  are  read  twelve  and  one-half  hun- 
dredths, and  thirty-three  and  one-third  hundredths,  respectively.  The 
former  may  be  written  .125. 

157.  Since  per  cent  means  hundredths,  in  reading  deci- 
mals we  may  say  per  cent  instead  of  hundredths.  The  symbol 
^  means  either  per  cent  or  hundredths. 

Thus,  .25  =  25  per  cent  =  25^.         .12i  =  12^  per  cent  =  Vit\%. 
.50  =  50  per  cent  =  50^.         .05  =  5  per  cent  =  5^, 

Read  the  following  mixed  numbers  : 


1. 

2.25. 

6. 

800.2035. 

11. 

136.00004. 

2. 

50.07. 

7. 

70.005. 

12. 

4000.004. 

3. 

13.033. 

8. 

30.078. 

13. 

3050.0507. 

4. 

310.09. 

9. 

3826.7. 

14. 

17005.017. 

5. 

7,394. 

10. 

4002.006, 

15, 

12345.12345, 

104  SCHOOL  ARITHMETIC. 

Read  both  ways  : 

16.  .15.  19.  .01.  22.  .37i.  25.  10^. 

17.  .27.  20.  .09.  23.  .80.  26.  35^. 

18.  .40.  21.  .75.  24.  .62^.  27.  33^^. 

Remark, — Decimals  may  be  read  in  different  ways.  Thus,  .125  may 
be  read  125  thousandths;  or  1  tenth,  2  hundredths,  5  thou.sandths;  or 
12  hundredths,  5  thousandths.  In  practice,  however,  it  is  desirable  to 
follow  the  method  indicated  in  the  Rule. 

WRITING    DECIMALS. 

158.  1.  Express  decimally  thirty -four  hundred -thou- 
sandths. 

Hundred-thousandths  is  the  fifth  decimal  order,  hence  5  places  are 
needed  to  express  the  decimal.  But  34,  written  as  an  integer,  occupies 
only  2  places,  leaving  3  places  on  the  left  to  be  filled  with  cipher's. 
Hence  the  decimal  is  written  .00034. 

l^  Pupils  should  become  thoroughly  familiar  with  the  names  of 
the  decimal  orders  at  least  to  millionths. 

Since  the  name  or  denomination  of  the  decimal  is  indi- 
cated by  the  position  of  the  right-hand  figure  with  respect 
to  ones'  place,  we  have  the  following 

Rule. — The  deiiomination  of  the  decimal  determines  the 
number  of  places  necessary  to  express  it;  therefore, 

Write  the  decimal  as  an  integral  iiumher,  and  prefix  ciphers, 
when  necessary,  to  supply  the  required  number  of  places, 
placing  the  decimal  point  directly  before  tenths. 

Express  decimally  : 

2.  Three  tenths.     One  tenth.     Nine  tenths.     Six  tenths. 

3.  Twelve  hundredths.  Four  hundredths.  Fifty-five  hun- 
dredths.    Ten  hundredths.     15^.     9  per  cent.     12|^. 

4.  Six  thousandths.  Fifteen  thousandths.  Two  hundred 
three  thousandths.  Twenty-five  ten-thousandths.  Seven 
ten-thousandths.     Four  hundred  fifty-one  ten-thousandths. 

6.  Eight  hundred-thousandths.   725  hundred-thousandths. 


FRACTIONS.  105 

5   millionths.      75   hundred-millionths.      351   thousandths. 
15  ten-millionths.     12  hundred-thousandths. 

6.  Four,  and  four  hundredths.  Three  hundred-thou- 
sandths. One,  and  five  thousandths.  Six  millionths.  Five 
hundred  thousandtlis. 

7.  Thirteen  hundredths.  47005  billionths.  Six  hundred, 
and  seven  thousandths.     Six  hundred  seven  thousandths. 

8.  Sixty-nine,  and  903  thousandths.  Forty-nine,  and  000 
ten-thousandths.     One  millionth. 

Write  in  words : 

9.  .7;  .05;  .01;  .016;  .203  ;  .25  ;  .324;  8^. 

10.  .001;  .0015;  .0125;  .2405;  .00025;  .00123. 

11.  2.7;  5.04;  6.008;  4.0010;  12.02301;  202.0202. 

12.  How  many  tenths  can  be  expressed  by  one  figure  ? 
How  is  10  tenths  written  ? 

13.  How  many  hundredths  can  be  expressed  by  one  figure  ? 
By  two  ?  How  is  100  hundredths  written  ?  How  is  100  per 
cent  Avritten  ? 

14.  Since  1  tenth  equals  10  hundredths,  100  thousandths, 
1000  ten-thousandths,  etc.,  it  is  plain  that  .1  =  .10  =  .100 
=  .1000  =  .10000,  etc.     Hence, 

159.  Principle. — Annexing  ciphers  to  a  decimal  reduces 
it  to  a  lower  denomination  without  changing  its  value. 

Query. — Does  omitting  ciphers  from  the  right  of  a  decimal  change 
its  value  ? 

1.  Change  .5,  .03,  .027,  and  .4850  to  thousandths. 

.5  =  .  500         In  the  first  two  decimals  we  annex  ciphers  enough 

.03       =:  .030  to    make    the    3    places   required     to   express    thou- 

.027     =  .027  sandths.     The  third  needs  no  changing.    Why  ?    The 

.4850  =  .485  last  is  changed  by  omitting  the  cipher  at  the  right. 

This  process  is  called  reducing  to  a  common  name  or  de- 
nomination (or  denominator). 

2.  Change  .8,  .25,  .030,  .4600,  and  .07  to  thousandths. 

3.  Reduce  .75,  .013,  .020,  and  .0146  to  ten- thousandtlis. 


106  SCHOOL  ARITHMETIC. 

4.  Reduce  .09,  .0240,  .3275,  .1,  .00010  to  millionths. 

5.  Change  .30,  5,  .400,  .8000,  and  1.7  to  tenths. 

6.  Change  .0032,  .2,  .470,  and  .835000  to  ten-thousandths. 

7.  Reduce  5  ones  to  tenths,  3  ones  to  hundredths,  10  ones 
to  tenths,  and  2  ones  toper  cent. 

8.  How  does  .1  compare  in  value  with  .01  ?  How  in  form  ? 
Then  what  is  the  effect  of  prefixing  a  cipher  to  .1? 

9.  How  does  .01  compare  in  vahie  with  .001  ?  Then  how 
is  .01  affected  hj  prefixing  a  cipher  ?  Prefix  ciphers  to  other 
decimals,  and  compare  values. 

160.  Pbinciple. — Prefixing  a  decimal  cipher  to  a  decimal 
divides  the  value  of  the  decimal  hy  ten. 

Queries. — 1.  How  does  prefixing  a  cipher  affect  the  place  of  each 
figure  in  the  decimal  ? 

2.  Does  a  figure  in  that  place  express  as  much  value  as  it  did  before 
being  moved  ? 

3.  What  part  of  its  former  value  does  it  express  ?  Then  by  what  has 
the  decimal  been  divided  ? 

ADDITION     AND     SUBTRACTION. 

161.  In  addition  and  subtraction  of  decimals  the  opera- 
tions are  the  same  as  the  like  operations  in  integral  numbers. 

1.  What  is  the  sum  of  .613,  .0176,  .2,  and  .601  ? 

•  613  By  arranging  the  decimal  points  in  a  vertical  line, 
.0176,  we  make  units  of  the  same  order  stand  in  the  same 

•  2  vertical  column.     The  numbers  are  added  precisely  as 
•oQl  in  integers,  and  the  decimal  point  is  placed  before 

1.4316  tenths.     (Is  14  tenths  a  fraction?    How  is  it  written?) 

2.  From  .3  subtract  .1235. 

(^)  (b)  By  arranging  the  decimal  points  in  a  verti- 

.6         =  .oOOO  cal  line,  we  cause  units  of  the  same  order  to 

•I^^^  ^^  .2235  stand  in  the  same  column.     We  subtract  as 

.1765        .1765  in  integers  or  U.  S.  money. 

Queries. — Why  may  .3  be  written  as  in  (b)  ?  (See  Art.  159.)  Is  it 
necessary  to  annex  ciphers  to  the  minuend  ?  When  the  rejuainder  is  q, 
mixed  puraber,  where  is  the  decimal  point  placecl  ? 


FRACTIONS.  107 

Find  the  value  of  the  following : 

3.  .17  +  .002  +  .2509.  11.  .75  -  .25. 

4.  .005  4-  .301  +  .29.  12.  .5  -  .005. 

6.  19.909  +  100.01  +  199.  13.  100.01-25.001. 

6.  .375  +  .048  +  255.0.  14.  10  -  .0678. 

7.  4.372  +  .4293  +  3.87.  15.  .10  -.06814. 

8.  5.0008  +  124  4-  .010.  16.  1000  -  .1000. 

9.  86.45  +  .001  +  .05.  17.  .6504  -  .067. 
10.  2.3  +  .004  +  .2  +  .88.  18.  .1  -  .0053. 

19.  94.61  +  .00421  +  .0003  +  .0044  +  10. 

20.  84.56  +  9.245  +  .8703  +  8.009  +  7.7. 

21.  1  million  —  one  millionth. 

22.  10.  —  10  ten-thousandths. 

23.  94  thousandths  —  253  ten-millionths. 

24.  25  thousandths  —  25  ten-thousandths. 

25.  1  —  1  thousandth  +  1  tenth  +  100  hundredths. 

26.  The  minuend  is  the  sum  of  .3  and  .003  ;  the  subtra- 
hend is  .02875.     What  is  the  remainder  ? 

27.  From  what  number  must  .0105  be  subtracted  to  leave 
the  remainder  1.807  ? 

28.  The  larger  of  two  numbers  is  3822.078  ;  their  differ- 
ence is  1934.124.     What  is  the  less  number  ? 

29.  Find  the  least  decimal  which  added  to  1.4142  —  .0022 
will  make  the  result  an  integer. 

30.  A  owes  11,000  to  B,  and  $1,347.55  to  C.  He  has  in 
cash  $1,955.75.  If  he  jmys  C  in  full,  how  much  will  he  lack 
of  having  enough  to  pay  B  ? 

31.  Mr.  Slaven  bought  an  organ  for  $85.50  on  a  credit  of 
three  months.  He  concluded  to  pay  cash,  and  was  allowed 
a  discount  of  $1.27.    How  much  had  he  left  out  of  a  $100  bill  ? 

32.  Bishop  Brothers  sold  goods  amounting  to  $190.50  on 
Monday,  $250  on  Tuesday,  $117.25  on  Wednesday,  $57  on 
Thursday,  $135.75  on  Friday,  and  $427.37  on  Saturday, 
What  were  the  total  sales  for  the  week  ? 


108  SCHOOL  ARITHMETIC. 

33.  Find  the  sum  of  345  millionths,  forty  and  40  millionths, 
seven  and  7  thousandths,  thirty-eight  and  87  ten-thou- 
sandths. 

34.  In  a  corncrib  that  will  hold  572.5  bushels  of  corn 
there  are  329.375  bushels.  How  many  bushels  will  be  re- 
quired to  fill  it  ? 

35.  One  side  of  a  square  field  is  42.375  rods  long.  If  12.5 
rods  of  the  fence  around  it  are  blown  down,  how  many  rods 
will  remain  standing  ? 

36.  A  tank  that  will  hold  1050.75  gallons  contains  396.7 
gallons.  If  135.5  gallons  be  added,  how  much  will  still  be 
needed  to  fill  the  tank  ? 

37.  A  man  bought  a  farm  for  $1,750  and  a  lot  for  $975. 75. 
For  what  amount  must  he  sell  both  to  gain  $289.50  ? 

MULTIPLICATION     AND     DIVISION. 

162.  The  processes  of  multiplication  and  division  of  deci- 
mals are  the  same  as  the  like  processes  in  integers,  the  locat- 
ing of  the  decimal  point  being  the  only  thing  that  needs 
special  attention. 

(a)  (b)  (c)  (d) 

.1  1.  .25  25 

1.  In  (a),  the  figure  1  expresses  1  tenth,  in  (b)  it  ex- 
presses 1  one.  What  has  been  the  effect  of  moving  the  deci- 
mal point  one  place  to  the  right  ? 

2.  In  (d),  the  2  expresses  2  ones,  in  (c)  2  tenths.  The 
5  in  (d)  expresses  5  tenths,  in  (c)  5  hundredths.  What  has 
been  the  effect  of  moving  the  decimal  point  one  place  to  the 
left  ? 

163.  Principle. — Each  removal  of  the  decimal  point  one 
place  to  the  right  multiplies  the  decimal  hy  10  ;  each  removal 
one  place  to  the  left  divides  the  decimal  hy  10. 

Thus,  by  moving  the  point  one  place  to  the  right,  .825  becomes  3.25  ; 
that  is,  3  tenths  have  become  3  ones,  the  2  hundredths  have  become  2 
tenths,  and  the  5  thousandths  have  become  5  hundredths.    Since  the 


FRACTIONS.  109 

value  of  each  figure  has  been  multiplied  by  10,  the  value  of  the  entire 
decimal  has  been  multiplied  by  10. 

IJ^"  Have  the  pupil  illustrate  the  second  part  of  the  principle,  which 
is  the  converse  of  the  first. 

164.  To  multiply  or  divide  a  decimal  by  lO,  lOO, 
lOOO,  etc. 

Rules. — 1.  To  multiply  a  decimal  hy  10,  100,  1000,  etc., 
move  the  decimal  point  as  many  places  to  the  right  as  there  are 
ciphers  in  the  multiplier,  annexing  ciphers  when  necessary. 

2.  To  divide  a  decimal  hy  10,  100,  1000,  etc.,  move  the  deci- 
mal point  as  many  places  to  theleft  as  there  are  ciphers  in  the 
divisor,  prefixing  ciphers  when  necessary. 

1.  Multiply  .275  by  100  ;  also  by  10000. 

.275  X  100  =  27.5.     .275  x  10000  =  2750. 

2.  Divide  .275  and  62.5  each  by  100. 


.275  -^ 

100  = 

.00275. 

62.5  H 

h  100 

=  .625. 

Find  the  value  of  : 

3.  3.25  X  10. 

8. 

37.5  ~ 

10. 

13. 

.37685  X  1000. 

4.  69.3  X  10. 

9. 

6.25  -^ 

10. 

14. 

52.16  X  1000. 

5.  .75  X  100. 

10. 

.314  ^ 

10. 

15. 

7.013  -7-  100. 

6.  .486  X  1000. 

11. 

.209  H- 

100. 

16. 

3875  -^  1000. 

7.  1.625  X  100. 

12. 

632  -^• 

100. 

17. 

41.065  -^  100. 

165.  To  multiply  or  divide  a  decimal  by  .1,  .Ol, 
.001,  etc. 

Rules. — 1.  To  multiply  hy  .1,  .01,  .001,  etc.,  move  the  deci- 
mal point  as  many  places  to  the  left  as  there  are  decimal  places 
in  the  multiplier. 

2.  To  divide  hy  .1,  .01,  .001,  etc.,  move  the  decimal  point  as 
many  places  to  the  right  as  there  are  decimal  places  in  the 
divisor. 

1.  Multiply  1.093  by  .1 ;  also  by  .01. 

1.093  X  .1  =  1.093  -J-  10  =  .1093. 
1.093  X  .01  =  1.093  --  100  =  .01093. 

To  multiply  a  number  by  .1  is  to  take  one  tenth  of   it ;  that  is,  to 


110  SCHOOL  ARITHMETIC. 

multiply  by  .1  is  to  divide  by  10  ;  to  multiply  by  .01  is  to  divide  by  100, 
etc.  By  comparing  the  products  with  the  multiplicands,  we  find  that  the 
decimal  point  has  been  moved  to  the  left  as  many  places  as  there  are 
ciphers  in  the  multiplier,     (See  Art.  164.) 

2.  Divide  32.5  by  .1  ;  also  by  .01. 

32.5  ~-  .1  =  32.5  X  10  =  325. 
32.5  -^  .01  =  32.5  X  100  =  3250. 

Since  there  are  10  tenths  in  1,  one  tenth  is  contained  in  any  number 
10  times  as  often  as  one  is  contained  in  it.  But  dividing  a  number  by  1 
does  not  alter  its  value.  Hence  to  divide  a  number  by  .1  is  to  multiply 
it  by  10  ;  to  divide  by  .01  is  to  multiply  by  100,  etc. 

Multiply  :  Divide  : 

3.  .258  by  10.  14.  37.5  by  100. 

4.  7.07  by  100.  15.  436  by  1000. 

5.  3.916  by  1000.  16.  .900  by  100. 

6.  .846  by  10000.  17.  24.57  by  1000. 

7.  7.5  by  .1.  18.  5  by  1000. 

8.  83.7  by  .01.  19.  .99  by  .1. 

9.  3.25  by  .01.  20.  .0075  by  .01. 

10.  .3004  by  .001.  21.  .0003  by  .001. 

11.  179.5  by  .001.  22.  4444  by  .0001. 

12.  3.428  by  .0001.  23.  18  by  .01. 

13.  .5  by  .0001.  24.  100  by  .1000. 

25.  Which  is  the  greater,  .5  x  100,  or  .5  -^  .01  ? 

26.  How  much  greater  is  .75  x  1000  than  .25  -^  .001  ? 

166.  To  niviltiply  or  divide  in  tlecimals — universal 
case. 

Principles. — 1.  Tlie  product  of  tioo  decimals  contains 
as  many  decimal  places  as  there  are  decimal  places  in  hoth 
factors. 

2.  The  quotient  of  two  decimals  contains  as  many  decimal 
places  as  the  number  of  decimal  places  in  the  dividend  exceeds 
the  number  in  the  divisor. 

The  number  of  decimal  places  in  the  dividend  can  be  increased  as  you 
please,  by  principle  in  Art.  159. 


FRACTIONS.  Ill 

1.  Multiply  .036  by  .27. 

036  The  multiplier   .27  =  27x  .01.     "We  therefore   multiply 

^27  first  by  27,  then  the  resulting  product  by  .01.  36  thousandths 
oKo  X  27  =  972  thousandths,  or  .972.  Multiplying  this  product 
wo  by  .01  moves  the  decimal  point  two  places  to  the  left  (Art. 
155).     Hence  the  required  product  is  .00972.     It  has  as  many 


00972 

decimal  places  as  both  factors  have. 

If  we  multiply  as  in  integers,  we  get  the  product  972,  to  which  we  pre- 
fix two  ciphers  to  make  the  required  five  places. 

2.  Divide  .00972  by  .27. 

.27).00972(.036        The  dividend  being  the  product  of  divisor  and 

81  quotient  must  contain  us  many  decimal    places  as 

]62  both  of  them.     Since  the  dividend  contains  5  decimal 

162  places  and  the  divisor  2,  the  quotient  must  contain  5 

—  2,  or  3  decimal  places.     Dividing  as  in  integers, 

we  get  the  quotient  36,  to  which  we  prefix  a  cipher  to  make  the  required 

three  places. 

167.  EuLES. — 1.  In  the  muUipUcation  of  decimals  multiply 
as  in  integers,  and  from  the  right  of  the  product  point  off  as 
many  decimal  places  as  there  are  in  hoth  factors,  prefixing 
ciphers,  if  necessary,  to  make  the  required  number  of  decimal 
places. 

2.  In  the  division  of  decimals,  divide  as  in  integers  (annex- 
ing ciphers,  if  necessary,  to  the  dividend),  and  point  off  from 
the  right  of  the  quotient  as  many  decimal  places  as  those  of  the 
dividend  exceed  those  of  the  divisor. 

If  the  quotient  does  not  contain  a  sufficient  number  of  decimal  places, 
ciphers  must  be  prefixed  to  make  the  required  number. 

(a)  Find  the  product  of  : 


1. 

.28  X  4.8. 

9. 

10000  X  .0001. 

2. 

.6  X  .7. 

10. 

7.5  X  .0005. 

3. 

.35  X  .16. 

11. 

1000000  X  .000001. 

4. 

10  X  .1. 

12. 

.001  X  10000. 

5. 

.134  X  25. 

13. 

.1  X  .1. 

6. 

216  X  .24. 

14. 

.5  X  .5. 

7. 

.478  X  .152. 

15. 

.5  X  .05. 

8. 

.0017  X  .09. 

16. 

.05  X  .005. 

112 


SCHOOL  ARITHMETIC. 


(a)  Find  the  product  of  : 

17.  .01  X  .001. 

18.  150  X  .1. 

19.  $1  X  .1. 

20.  7  X  1.1. 

21.  2.5  X  2.5. 

22.  $100  X  .06. 

23.  .017  X  3.7. 

24.  101  X  1.01. 

25.  1.03  X  1.09. 

26.  5.005  X  .005. 

(b)  Find  the  quotient  of 

1.  .00125  -T-.5. 

2.  .0075-^1.5. 

3.  1  --  .1. 

4.  .01  ^  100. 

5.  16.84  -f-  .02. 

6.  .00884 -^  .34.      . 

7.  .0355  ^  .71. 

8.  16.025  -^  .045. 

9.  10000  -^  .0001. 

10.  .000375  -^  .0005. 

11.  1000000  -f-  .000001. 

12.  .000001  -^  1000000. 

13.  1150  ^  $  .06 

14.  II  -^  $  .05. 

15.  159.750^  .00375. 

16.  14400-^.32. 

17.  14400  ^  3.2. 

18.  200  -^  .002. 

19.  .735  -J-  500. 

20.  78.13^5. 

21.  78.39 -f- 3. 

22.  125  ~  25000. 


27.  .008  X  800. 

28.  5    tenths    x    50   hun- 

dredths. 

29.  .01  X  .1  X  1. 

30.  .05  X  5  X  .50. 

31.  72.5  X  10. 

32.  .1225  X  .1. 

33.  25.6  X  .20. 

34.  .054  X  100. 

35.  125  X  1.05. 


23.  12  -=-  .0012. 

24.  5.4768-^22.82. 

25.  .025-4-  250. 

26.  .0567  -^  43. 

27.  1  -i-  3.1416. 

28.  ten  -=-  .01. 

29.  1  millionth -f-.  01. 

30.  300  hundredths  ~  15 

tenths. 

31.  3.1416  -^  .31416. 

32.  .25  -^  .0025. 

33.  9  ones  -^  40  tenths. 

34.  25    tenths  -^  25    hun- 

dredths. 

35.  25  hundredths  ~-  .025. 

36.  27.45  -r  1.5. 

37.  250  ~-  .025. 

38.  2750  -4-  .25. 

39.  3.609  -4-  .9. 

40.  27.63 -^  .003. 

41.  4.914-4-  70. 

42.  .026  -^-  .000013. 


FRACTIONS.  .  113 

108.  1.  Multijily  4.Ge5  by  700,  and  divide  the  product  by 
300. 

4.G5  X  100  =  4r,5.     Then  405  x  7  =  3255. 
3255  -^  100  =  32.55  ;  and  32.55  -f-  3  =  10.85. 

Find  tlie  vahie  of  : 

2.  52(3.53  X  50.  7.  030  -^  500. 

3.  245.0  X  400.  8.  .844^  400. 

4.  .804  X  900.  9.  307.2  -i-  1200. 
6.  .7854  X  700.  10.  2607.5  -^  8300. 
6.  150  X  25^.  11.  150  -^  25^. 

12.  What  is  the  value  of  .05  x  .07  +  .28  ^  .5  ? 

13.  If  I  give  3  i:>igs  for  17.50,  how  many  must  I  give  for 
$37.50? 

14.  A   man  paid  $17.25  for  300  pounds  of  sugar.     What 
did  it  cost  per  pound  ? 

15.  How  many  eggs  in  a  crate  containing  24.5  dozen  ? 

16.  If  8  pounds  of  coffee  cost  $1.74,  what  will  5  pounds 
cost  ? 

17.  At  2|^  each,  how  much  will  3.25  dozen  lemons  cost  ? 

18.  A  man  paid  $15  for  rice,  at  the  rate  of  4  pounds  for  a 
quarter.     How  many  pounds  did  he  get  ? 

19.  A  has  $1.40  and  B  has  2.5  times  as  much.     How  much 
must  B  give  A  so  that  each  may  have  the  same  amount  ? 


BILL.S   AND   ACCOUNTS. 


169.  Prof.  Samuel  Andrews  bought  of  J.  R.  Weldiii  & 
Co.  the  following  :  6  dozen  lead  pencils  at  1.30  a  dozen,  2 
gross  pens  at  $.85,  5  reams  note  paper  at  11.50,  and  20 
arithmetics  at  $.75. 

In  a  few  days  he  received  the  following  hill  : 

Columbia,  S.  C,  May  1,  1899. 
Mr.  Samuel  Andrews, 

Bought  of  J.  R.  Weldin  &  Co. 


To  6  dozen  Lead  Pencils  @    $.30 

1 

80 

''   2  gross  Pens                  ''       .85 

1 

70 

''  5  reams  Note  Paper     ''     1.50 

7 

50 

"   20  Arithmetics             ''       .75 

15 

00 

26 

00 

When   this  bill  was  paid,  the   following  was  written   on 
it  as  a  receipt : 

*^  Received  payment, 

J.  R.  Weldin  &  Co. 

(The  "  G  "  is  the  initial  of  Mr.  Greene,  who  receipted  the  bill.) 

1.  Mrs.  R.  D.  White   ordered  the  following  from  Davis 
&  Russell,  New  Orleans,  La.  : 

18  yd.  Scotch  Gingham  @  21^. 

36i  yd.   Calico  @  6^. 

12i  yd.  India  Silk  @  45^. 

25  yd.  Cashmere  @  $1.25. 
Make  out  her  bill,  and  receipt  it. 


BILLS   AND  ACCOUNTS.. 


115 


170.  The  following  is  a  specimen  of  a  receipted  bill,  with 
a  discount,  and  credits  : 

Jackson,  Miss.,  Oct.  1,  1899. 
Mr.  T.  B.  DeArmit, 

To  Gordon,  Hay  &  Co.,  Dr. 


1899. 
Jan.  13 
May  21 

Aug.    9 

To  50  Grammars...  $.40 
'*    24  Arithmetics. .    .60 
'*    42  Histories....   1.00 

Less  10^ 

20 
14 
42 

00 
40 
00 

76 

7 

40 

64 

Cr, 

By  Cash $25.00 

"        '^    25.00 

July  25 
Sept.    8 

68 
50 

76 

00 

18 

76 

Received  payment, 

Gordon,  Hay  &  Co. 
By  WiLSOK. 

171.  A  Debt  is  the  amount  which  one  person  owes  another. 
A  Debtor  is  a  person  or  firm  that  owes  a  debt. 

172.  A  Credit  is  the  amount  paid  on  a  debt.  A  Cred- 
itor is  a  person  or  firm  to  whom  a  debt  is  due. 

In  the  transaction  mentioned  in  Art.  170,  who  is  the 
debtor  ?     Who  is  the  creditor  ?     Name  the  credits. 

173.  An  Account  is  a  record  of  the  debts  and  credits  be- 
tween two  parties — a  debtor  and  a  creditor. 

174.  A  Bill  is  a  creditor's  written  statement  of  the  quan- 
tity and  price  of  each  item  in  liis  account  with  a  debtor, 
together  with  the  discount  and  credits,  if  any,  and  the  net 
amount  due. 

Bills  are  commonly  called  invoices. 


116  SCHOOL  ARITHMETIC. 

175.  A  Statement  is  a  written  summary  of  an  account 
between  two  parties,  rendered  at  stated  intervals,  usually 
monthly. 

1  76.  Make  out  and  receipt  the  following  bills.  Supply 
dates  and  names  where  needed. 

1.  Mr.  R.  P.  Lougeay  bought  of  McAllister  &  Co.  25  pounds 
of  coffee  at  28  cents  a  pound,  75  pounds  of  sugar  at  5^  cents 
a  pound,  and  20  pounds  of  prunes  at  12  cents  a  pound. 

2.  Mrs.  M.  B.  Kifer  bought  of  Campbell  &  Smith  10  yards 
of  silk  at  11.50  a  yard,  36  yards  of  muslin  at  7  cents  a  yard, 
15  yards  of  flannel  at  $.75  a  yard,  jind  2  pairs  of  shoes  at 
$3.25  a  pair. 

3.  Mrs.  A.  C.  McLean  bought  of  Kauffman  Bros.  3  table- 
cloths @  $3.50,  1  piano  cover  @  $4.75,  4  pairs  of  lace  cur- 
tains @  $5.25,  2  doz.  towels  @  $3.60  a  dozen,  and  12  yards 
cashmere  @  $1.25  a  yard. 

4.  Miss  Xiinnie  Mackrell  bought  of  W.  M.  Laird  2  pairs 
ladies'  shoes  @  $2.75  a  pair,  6  pairs  overshoes  @  $.75  a 
pair,  1  pair  Oxford  ties  @  $1.25,  3  pairs  misses'  shoes  @ 
$2.15,  and  1  pair  gum  boots  @  $3.25. 

5.  Mr.  S.  M.  Brinton  bought  of  Hopper  Bros.  &  Co.  2 
doz.  silver  knives  @  $36  a  dozen,  4  doz.  silver  teaspoons  @ 
$16  a  dozen,  2  doz.  silver  tablespoons  @  $10.25  a  set,  and  1 
silver  spoon  bolder  for  $9. 

6.  Mr.  Wm.  Hasley  bought  of  Fred  Gray  12.5  tons  of  coal 
@  $3.25,  40  bushels  of  apples  @  $.75,  200  lb.  grapes  @  3 
cents  a  pound,  and  25  bushels  of  potatoes  @  $.85. 

7.  On  May  25,  J.  M.  Logan  bought  of  W.  II.  Keech  5 
bedsteads  @  $14,  1  bookcase  for  $35,  and  18  chairs  @  $15  a 
dozen.  On  July  3,  he  bought  3  hammocks  @  $2.25,  and  a 
leather  couch  for  $45.  On  June  15,  he  paid  $50  in  cash,  and 
on  the  10th  $37.50  more. 


REVIEW   WORK. 


ORAL    EXERCISES. 


177.  1.  How  Tiijiny  tenths  in  80  hundredths  ? 

2.  How  many  hundredths  in  7  tenths  and  15  liundredtlis  ? 

3.  I  paid  3  tenths  of  a  dollar  for  3  cakes  of  soap.  At  the 
same  rate,  how  much  would  I  pay  for  a  dozen  cakes  ? 

4.  A  lady  spent  .1  of  her  money  for  a  hat,  and  A  for  a 
shawl,  and  the  remainder  for  a  dress  which  cost  l?15.  How 
much  had  she  at  first  ? 

5.  At  $.09  each,  how  many  slates  can  be  bought  for  $3.00  ? 

6.  If  80  is  divided  into  10  ecjual  parts,  what  is  one  jiart 
called?  Three  parts  ?  Nine  parts  ?  How  many  are  7  tenths 
(.7)  of  80? 

7.  If  18  is  .3  of  some  number,  what  is  the  number  ? 

8.  Thirty-five  is  .5  of  what  number  ? 

9.  Of  wliat  nnmber  is  9  three  tenths  ? 

10.  The  sum  of  .2  and  .05  is  .5  of  what  number  ? 

11.  A  has  $1.50,  and  .3  of  his  money  is  .1  of  B's  money. 
How  much  has  B  ? 

12.  B  and  C  together  have  $40.  If  .3  of  B's  money  equals 
.9  of  C's,  how  much  has  each  ? 

13.  How  often  must  .3  be  added  to  itself  to  make  3  ? 

14.  How  many  times  must  .7  be  subtracted  from  3.5  to 
leave  a  remainder  of  1.4  ? 

15.  How  many  hundredths  can  be  taken  from  25  tenths  ? 

WRITTEN    EXERCISES. 

178.  1.  If  seven  sheep  are  worth  $31.50,  how  many  sheep 
can  be  bought  for  $184.50  ? 


118  SCHOOL  ARITHMETIC. 

2.  A  man  divided  his  farm  of  227.5  acres  into  14  equal 
fields.     How  many  acres  in  5  of  the  fields  ? 

3.  At  $2,625  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $55,125  ? 

4.  If  a  person's  taxes  are  5.8  mills  on  $1,  how  much  will 
they  be  on  $2500  ? 

5.  Find  the  cost  of  237.25  bushels  of  oats  at  .42  of  a 
dollar  a  bushel. 

6.  At  $.08  each,  how  many  copy  books  can  be  bought  for 
$24? 

7.  Gold  weighs  19.36  times  as  much  as  an  equal  bulk  of 
water,  and  a  cubic  foot  of  water  weighs  62.5  pounds.  How 
many  cubic  feet  of  gold  weigh  a  ton,  or  2,000  pounds  ? 

8.  One  pound  of  dry  oak'  wood  when  burnt  yields  .023  of 
a  pound  of  ashes.  How  many  pounds  must  be  burnt  to  pro= 
duce  46  pounds  of  ashes  ? 

9.  Every  day  a  newsboy  buys  70  papers  at  30  cents  a 
dozen,  and  sells  them  at  5  cents  each.  How  much  money 
does  he  make  in  6  days  if  40  papers  remain  unsold  ? 

10.  If  a  boy  saves  6  dimes  a  week,  in  how  many  days  can 
he  save  enough  to  buy  a  suit  worth  $5.40  ? 

11.  How  often  can  .013  be  subtracted  from  26  ? 

12.  Find  the  cost  of  8  bushels  3  pecks  of  turnips  at  $.125  a 
peck. 

13.  The  divisor  is  27.125,  the  quotient  7.32,  and  the  re- 
mainder 18.0825.     What  is  the  dividend  ? 

14.  Divide  3  ten-millionths  by  10  millionths,  and  multiply 
the  quotient  by  30. 

15.  At  $1.50  a  thousand,  what  will  1,750  envelopes  cost  ? 

16.  The  distance  around  a  circle  is  about  3.1416  times 
the  distance  across  it  through  the  center.  If  the  distance 
around  a  circular  pond  is  50  feet,  what  is  the  distance  across 
it? 

17.  Two  men  start  from  the  same  place  at  the  same  time 
and  travel  in  the  same   direction,  one  going  3.28  miles  an 


REVIEW  WORK.  119 

hour,  the  other  4.07  miles  an  hour.    How  far  apart  will  they 
be  in  9  hours  ? 

18.  The  circumference  of  the  wheel  of  a  bicycle  is  11.28 
feet.  How  many  times  will  it  turn  in  going  2.5  miles,  there 
being  5280  feet  in  a  mile  ? 

19.  Find  the  cost  of  8375  feet  of  lumber,  when  lumber  is 
worth  $18  a  thousand  feet. 

20.  In  a  city  of  240,000  inhabitants  .125  of  tlie  population 
are  school  children.  If  each  teacher  has  50  pupils,  how 
many  teachers  are  in  that  city  ? 

21.  Add  155  ones,  155  tenths,  155  hundredths,  155 
thousandths. 

22.  The  divisor  5.125  is  5  times  the  quotient;  what  is  the 
dividend  ? 

23.  The  product  of  three  factors  is  78. GG  ;  two  of  the 
factors  are  respectively  6.9  and  7.125;  what  is  the  third 
factor  ? 

24.  Find  the  least  decimal  fraction  which  added  to  the 
sum  of  87.43  and  1G9.578  will  make  the  sum  an  integer. 

25.  A  man  paid  .15  of  his  money  for  rent,  .02  for  wood, 
.18  for  clothing,  and  had  $812.50  left.  How  much  had  he  at 
first? 

26.  Find  the  product  of  the  two  smallest  decimals  that  can 
be  expressed  by  the  figures  0,  0,  9,  and  3. 

27.  Gunpowder  is  composed  of  .76  nitre,  .14  charcoal,  and 
.10  sulphur.  How  much  of  each  is  required  to  make  2000 
pounds  of  powder  ? 

28.  At  $0.34  a  bushel,  how  many  barrels  of  apples  can  be 
had  for  $13.60,  allowing  2.5  bushels  to  the  barrel  ? 

29.  How  many  pounds  of  butter  could  be  made  from  46 
cows  during  the  month  of  June,  each  cow  averaging  2.5 
gallons  of  milk  daily,  and  each  gallon  making  .5  of  a  pound 
of  butter  ? 

30.  If  4  cords  of  wood  are  worth  as  much  as  13.4  bushels 
of  rye,  how  much  rye  can  be  obtained  for  15  cords  of  wood  ? 


120  SCHOOL  ARITHMETIC. 

31.  If  a  Mexican  dollar  is  worth  10.85,  how  many  Mexican 
dollars  equal  the  value  of  |G80  in  U,  S.  money  ? 

32.  If  the  land  that  produces  a  bale  of  cotton  yields  30 
bushels  of  cotton  seed,  what  is  the  value  @  $.30  per  bushel  of 
the  cotton  seed  produced  by  the  land  that  yields  21  bales  of 
cotton  ? 

33.  In  one  manufacturing  establishment  the  average  weekly 
wages  paid  to  2()2  operators  was  $12.85  ;  in  another,  to  355 
operators,  $13.84;  and  in  a  third,  to  128  operators,  $15.11. 
Find  the  average  weekly  wages  in  all  three  establishments. 

34.  If  a  railroad  train  runs  350  miles  in  19.5  hours,  but 
makes  three  stops  of  20  minutes  each,  and  ten  stops  of  (5 
minutes  each,  what  is  the  average  rate  per  hour  while  run- 
ning ? 

36.  A  franc  is  19.3  cents.  Find  the  cost  in  United  States 
money  of  goods  bought  in  Paris  amounting  to  1,000  francs. 

36.  A  cubic  foot  of  water  weighs  1000  ounces.  IIow 
many  pounds  does  a  cubic  foot  of  gold  weigh,  gold  being  19.4 
times  as  heavy  as  water  ? 

37.  If  oysters  yield  1.25  gallons  to  the  bushel,  how  many 
bushels  in  the  shell  must  I  buy  so  that  when  opened  they  will 
fill  a  10-gallon  can  ? 

38.  In  the  year  1897,  the  total  ordinary  expenditures  of  the 
United  States  government  were  $365,774,159,  which  was  $5.02 
to  each  person.  What  was  the  population  in  that  year,  to 
the  nearest  1000  ? 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

179.  1.  Can  75  tenths  be  written  as  a  decimal  fraction  ? 
Why  not  ?     Can  it  be  written  as  a  mixed  number  ? 

2.  Divide  one  by  seven,  carrying  the  quotient  to  12  decimal 
places,  and  carefully  note  the  result. 

3.  Investigate  the  result  of  dividing  one  by  3,  11,  13,  and 
17,  carrying  the  division  as  far  as  may  be  necessary. 


REVIEW  Work.  121 

4.  After  spending  .015  of  liis  mouey,  and  2  tenths  of  the 
remainder,  B  luid  $15.70  left.      How  much  did  he  spend  ? 

5.  Find  the  cost,  at  *S  75  per  thoiisund,  of  the  rails  for  040 
panels  of  14- rail  fence. 

6.  The  nniliipliciand  is  .005,  and  tlie  product  is  1.  hy 
what  must  the  multi[)lier  he  divided  to  give  a  quotient  ecjual 
to  the  product  ? 

7.  I  gave  .44  of  my  money  for  a  fjirm,  and  .75  of  the  re- 
mainder for  a  store.  If  the  farm  cost  ^250  more  than  the 
store,  how  much  did  I  pay  for  the  store  ? 

8.  Cork  weighs  15  pounds  per  cubic  foot,  and  its  weight  is 
.24  of  the  weiglit  of  water.  Find  tiie  weight  of  10  cubic  feet 
of  oak,  if  the  weight  of  oak  is  .934  of  the  weight  of  water. 

9.  The  distance  of  the  moon  from  the  earth  is  50.07  times 
tlie  earth's  radius.  If  this  radius  is  30()2..S24  miles,  find  the 
distance  to  the  moon. 

10.  In  1800  the  native  population  of  the  United  States  was 
85.23^  of  tlie  whole.  Tlie  foreign  born  was  what  decimal 
part  of  the  native  ? 

11.  The  population  of  Italy  is  29,090,785.  The  total  in- 
debtedness of  the  country  is  $2,324,826,329.  Find  tlic  rate 
of  debt  for  each  person. 

12.  The  population  of  New  Orleans  in  1807  was  280,000. 
The  assessed  valuation  of  taxable  property  was  $140,054,475. 
Supposing  the  whole  population  to  be  divided  into  families 
of  five,  compute  the  average  wealth  of  each  family. 


COMMON   FRACTIONS. 

180.  1.  When  anything  is  divided  into  ten  equal  parts, 
what  is  one  part  called  ?  What  name  is  given  to  tenth-parts 
of  anything  ?^  What  kind  of  fraction  is  one  expressing  deci- 
mal parts  of  any  whole  or  quantity  ?  Then  what  kind  of 
fraction  is  one-tenth  (.1)  ? 

2.  When  anything  is  divided  into  100  equal  parts,  what  is 
one  part  called  ?  25  parts  ?  What  kind  of  fraction  is  6  hun- 
dredths ?     Fifty  thousandths  ?     How  are  they  written  ? 

3.  Is  one  fourth  yard  a  decimal  fraction  ?  $4  ?  5  half- 
hushels  9     Why  not  ? 

181.  When  a  measuring  unit  is  a  decimal  part  of  a  certain 
whole  or  quantity,  any  number  of  these  units  (parts)  ex- 
pressed in  decimal  notation  is  called  a  decimal  fraction  ;  but 
when  the  Unit  of  measure  is  one  of  any  number  of  equal 
parts,  one  or  more  of  such  parts,  expressed  by  two  numbers 
one  above  the  other  with  a  line  between  them,  is  called  a 
Common  Fraction. 

Thus,  5  tenths  (.5)  and  25  hundredths  (.25)  are  decimal  fractions,  while 
3  fourths  (f)  and  10  thirds  (^3^)  are  common  fractions. 

Notes. — 1.  The  decimal  point  was  first  used  about  the  beginning  of  the 
17th  century,  but  100  years  elapsed  before  decimal  fractions  were  exten- 
sively employed. 

2.  During  the  introduction  of  this  new  class  of  fractions,  the  "old 
style"  fractions  were  given  the  name  "vulgar "or  "common"  frac- 
tions to  distinguish  them  from  the  others,  which  were  written  in  a  new 
and  special  way  by  the  aid  of  the  decimal  point. 

3.  Strictly  speaking,  a  number  of  decimal  parts,  to  be  called  a  deci- 
mal fraction,  must  be  written  in  the  form  peculiar  to  decimals,  since  it 
is  only  in  that  form  that  the  practices  exemplified  in  decimals  are 
applicable. 


FRACTIONS.  123 

182.  The  Unit  in  fractions  is  v^part  of  one  whole,  and  is 
called  a  Fractional  Unit;  that  is,  a  fourth  is  the  unit  of 
measure  in  three  fourths. 

183.  The  Denominator  (nanier),  the  number  written 
below  the  line,  shows  the  number  of  equal  parts  into  which 
a  certain  whole  or  quantity  is  divided  in  order  to  obtaisi  the 
fractional  unit. 

Thus,  in  f  bushels,  the  denominator  2  shows  that  the  one  bushel  is 
divided  into  two  equal  parts  and  the  unit  is  a  half-bushel.  How  many- 
fractional  units  are  used  to  express  the  value  of  the  quantity  ? 

184.  The  Numerator  {nnmberer),  the  number  written 
ahove  the  line,  shows  how  many  of  the  fractional  units  are  used. 

Queries. — In  the  fraction  $},  which  is  the  denominator  ?  What 
does  it  show  ?  What  does  the  3  show  ?  Of  what  does  %l  express  the 
value  ? 

Note. — A  fraction  whose  numerator  is  less  than  the  denominator  is 
called  a  proper  fraction  ;  otherwise  it  is  usually  called  an  improper  frac- 
tion. 

185.  A  Mixed  Number  is  the  sum  of  an  integer  and  a 
fraction,  and  is  expressed  by  writing  the  fraction  imme- 
diately after  the  whole  number. 

Thus,  $5  +  If  =  $5|,  is  a  mixed  number. 

Notes. — 1.  The  integer  and  fraction  must  be  like  numbers, 

2.  Five  silver  quarters  are  equivalent  to  $li  ;  but  in  this  case  the 
mixed  number  is  7iot  the  sura  of  an  integer  and  a  fraction.  The  quan- 
tity is  ?k  fraction  ($|)  made  up  of  5. units  of  measure  (quarters).  In  prac- 
tice, however,  no  attention  is  paid  to  this  distinction. 

3.  The  definition  of  a  fraction  in  Art.  181  includes  only  fractions  whose 

2  5 

denominators  are  integral,  hence  excludes  such  expressions  as  — -  and  — -, 

which  are  properly  expressions  of  unexecuted  division.  (See  Art.  95.) 
However,  in  algebra  the  definition  of  a  fraction  is  extended  so  as  to  em- 
brace any  expression  in  the  fractional  form. 

186.  An  Integer  or  a  Mixed  Numler  may  be  expressed 
in  the  form  of  a  fraction,  and  treated  as  a  fraction. 

Thus,  |4  =  1^  =  4  one  dollars  ;  $3^  =  $|  =  7  half-dollars  ;  1  =  i  = 
l=ietc. 


124  SCHOOL  ARITHMETIC. 

187.  The  number  of  equal  parts  into  which  anytliing  la 
divided  gives  the  parts  their  name. 

Thus,  any  quantity  divided  into  2  equal  parts  =  2  lialves,  or  J; 
any  quantity  divided  into  3  equal  parts  =  3  thirds,  or  ^ ; 
any  quantity  divided  into  4  equal  parts  =  4  fourths,  or  J; 
any  quantity  divided  into  5  equal  parts  =  5  fifths,  or  §  ; 
etc. 

Therefore,  to  read  fractions. 

Direction. — State  the  numher  of  fractional  units  (units 
of  measure)  and  give  tlteyn  the  name  indicated  by  the  denomi- 
nator. 

(a)  Read,  and  write  in  words  : 


1. 

h 

1. 

!• 

4. 

U>   M.   If 

7. 

u, 

:.'„'6.     H- 

2. 

h 

i> 

*• 

5. 

ih  ih   itV- 

8. 

M. 

,%%,     .42. 

3. 

i 

■1. 

A- 

6. 

-?\«>    ih'    /A- 

9. 

?' 

1'     ««• 

(b)  Write  in  figures  : 

1.  Three  fourths.  7.  Fifty-two  hundred tlis. 

2.  Four  fifths.  8.  Sixty  two-hundredths. 

3.  Ten  elevenths.  9.  Nine  eightieths. 

4.  Eight  twenty-firsts.  10.  Eighty  ninetieths. 

5.  Twelve  thirty-thirds,  11.  Forty  three-thousandths. 

6.  Two  hundredths.  12.  Two  and  one-half  thirds. 

CHANGE     OF    FORM. 

188.  To  change  fractions  to  larger  denominators. 

1.  How  many  fourths  in  a  gallon  ?  In  ^  of  a  gallon  ? 
Then  ^  is  equal  to  how  many  fourths  ? 

2.  How  many  eighths  in  an  apple  ?  lu  ^  of  an  apple  ? 
Then  ^  is  equal  to  how  many  eighths  ? 

3.  Since  ^^  =  |,  and  I  —  g,  what  has  been  done  to  the 
numerator  and  denominator  of  tlie  fraction  |  without  chang- 
ing its  value  ?     How  has  it  been  done  ? 

4.  Change  ^  to  sixths.    To  twelfths.    To  16ths.    To  20ths. 


FRACTIONS.  125 

5.  Change  §  to  sixtlis.     To  niiitlis.     To  12tlis.     To  ISths. 

6.  Ill  1  liow  many  24ths  ?  Then  how  many  24ths  are  there 
in  ^  ?     In  ^  ?     In  f  ?     In  j\  ? 

7.  Name  three  fractions  each  equal  to  J. 

8.  What  may  be  done  to  the  numerator  and  denominator 
of  a  fraction  without  changing  its  value  ? 

9.  Express  I  with  nunieratorand  denominator  4  times  as 
large,  and  explain  why  the  value  of  the  fraction  is  not 
changed. 

Multiplying  numerator  and  denominator  by  4,  we  have  |f .  Since  the 
fraclional  units  are  ^  as  large,  while  the  number  of  them  taken  is  4 
times  as  great,  jg  is  equivalent  to  f .     Hence  the  following 

189.  Principle. — Multiplying  numerator  and  denomina- 
tor hy  the  same  number  does  not  change  the  value  of  the  frac- 
tion. 

WRITTEN     EXERCISES. 

190.  1.  Change  |  to  twelfths. 

j2  _i_  3  :::::   4  Multiplying  numerator  and  denominator  by  the  same 

2        j^         o      number  does  not  change  the  value.    Since  the  denomina- 

—  =  —     tor  is  to  be  12,  both  numerator  and  denominator  must 

3x4        12     be  multiplied  by  12  -^  3,  or  4  ;  therefore,  f  =  h- 
Analysis. — In  |,  or  one-  there  are  12  twelfths,  hence  in  \  there  are  J 

of  12  twelfths,  or  4  twelfths,  and  in  f  there  are  2  times  4  twelfths,  or  ]*2. 

Direction. — Multiply  numerator  and  denominator  hy  the 
quotient  arising  from  the  division  of  the  required  denomina- 
tor hy  the  given  denominator. 

Change 

^'  h  3'  i'    ''^^^^   i  ^^   twelfths. 

3.  ^,  i,  |,    and.  -^^   to   twenty-fourths. 

4.  1,  f,  -^Q,    and   ^f  to   fortieths. 
5-  3,  f,  J,   ^,    and   ^  to   36ths. 

6.  h   h   /o.    ih   and    A   to   lOOths. 

7.  A,   j\.    1%   ih    and   ^  to  144ths. 
8-  f   h   t\,   fj,   and   U  to   385ths. 


126  SCHOOL   ARITHMETIC. 

191.  To  change  fractions  to  smaller  denominators. 

1.  How  many  twelfths  in  1  ?  How  many  halves  ?  Then 
12  twelfths  =  how  many  halves  ? 

2.  How  many  halves  in  6  twelfths  ? 

3.  How  many  thirds  in  1  ?  Then  how  many  thirds  in  12 
twelfths  ?    In  4  twelfths  ?    In  eight  12ths  ? 

4.  Since  A  ==  i,  and  -^  =  i,  what  has  been  done  to  the 
numerator  and  denominator  of  the  fractions  -j%  and  -^^  with- 
out changing  the  values  ?     How  has  it  been  done  ? 

5.  Change  j\  to  fourths,     -f^,  \\,  \%. 

6.  How  many  fourths  in  f  ?  Is  f  =  f  ?  What  change  has 
been  made  in  the  denominator  ?  What  corresponding  ghange 
in  the  numerator  ? 

Dividing  numerator  and  denominator  by  3,  we  have  \.  Since  the 
fractional  units  are  twice  as  large,  while  the  number  of  them  is  ^  as 
great,  f  is  equivalent  to  f .      Hence  the  following 

192.  Principle. — Dividing  denominator  and  numerator 
of  a  fraction  hy  the  same  number  does  not  change  the  vahie 
of  the  fraction. 

WRITTEN    EXERCISES. 

193.  1.  Change  ^}  to  twelfths. 

Dividing  th^  given   denominator 
24  -h  12  =:     2  by  the  required  denominator  we  get 

16  -i-     2 8  2,   which  must    be  used   as  divisor 

24  _^     2        12  ^^    ^^^^    numerator    and    denomi- 

nator. 
Queries. — 1.  In  this  process,  how  has  the  size  of  the  fractional  unit 
been  changed  ?    2.  How  correspondingly  has  the  number  of  fractional 
units  been  changed  ?    3.  Which  gives  more  definite  idea  of  the  value  of 
the  quantity,  ||  yd.  or  |  yd,  ? 

Change  to  smaller  denominators : 

4.  if;f*;a;«;4f;«;«. 

Queries. — (a)  When  should  you  express  a  fraction  with  smaller  denomi- 


FRACTIONS.  127 

nator  ?    (b)  Can  ,^2  l>e  changed  to  smaller  denominator  without  a  change 
of  value  ?    Can  J  ?    Can  ^  ?    Are  2  and  3  prime  to  each  other  ?    Why  ? 

194.  A  fraction  is  expressed  with  its  stnallest  denominator 
when  its  numerator  and  denominator  are  prime  to  each  other. 

WRITTEN    EXERCISES. 

195.  1.  Change  }f  to  its  smallest  denominator. 
30  -^  3  =  10 

45  -^  o  =  10  Divide  numerator  and  denominator 

Therefore,  by  3,  then  by  5.     Since  no  number  will 

f  ^  —  il*  exactly  divide  both  numerator  and  de- 

10  -T-  5  =     2  nominator  of  the  fraction  f,  the  frac- 

15  -4-  5  =    3  tion  is  changed,  or  reduced,  to  its  small- 

Therefore,  «*^  denominator. 

IS  =  I- 

Direction. — Divide  numerator  and  denominator  by  com- 
mon factors  successively  until  they  are  prime  to  each  other. 
Change  to  smallest  denominators  : 

14.  HI.  m.  m- 

15.  iM.  m,  m- 
16-  ttf.  f«.  m- 

17.  m,  m>  m- 

18.  m,  Mf.  m- 
19-  fh>  1%.  m- 

196.  To  change  integers  or  mixed  numbers  to  frac- 
tional form,  and  the  reverse. 

1.  How  many  quarter-dollars  in  one  dollar  ? 

2.  How  many  fourths  in  a  dollar  ?     In  2  dollars  ?     In  2^ 
dollars  ?     In  2f  apples  ?     Illustrate  with  objects. 

3.  How  many  apples  have  you  when  you  have  eight  quar- 
ters, or  fourths  ?     Ten  fourths  ?     17  fourths  ? 

4.  How  many  eigliths  in  1  ?    In  2  ?    In  3  ?     In  5  ?    In  2f  ? 
In  3f  ? 

5.  How  many  l^s  in  eight  eighths  ?    In  16  eighths  ?    In 
Y  ?     In  ^  ?     In  1/  ?     In  ^  ? 


2.  if.  if,  H- 

8. 

T%,  m,  iVt- 

3.  n,  ft,  !%• 

9. 

IM,  Hi  IM- 

4.  If,  li  41- 

10. 

ili,  iM,  iJi- 

5.  ft,  if,  -ji 

11. 

Mi,  ^,  m- 

6.  «,  15,  It- 

12. 

m,  tVV  iH- 

7.  M,  li  M. 

13. 

Ul.  iff,  ili- 

128  SCHOOL  ARITHMETIC. 

6.  Twenty-five  half-dollars  are  how  mauy  dollars  ? 

7.  How  many  quarters  will  pay  for  a  pig  that  costs  $2^  ? 

8.  How  many  fractional  units,  thirds,  are  there  in  3^  ? 
In  5|  ? 

9.  How   many   ones   are   there    in    ^jt  p     j^  s^_  ?     in  ^  ? 
In  4j9  ?     In  V  ?     I"  V  ? 

WRITTEN     EXERCISES. 

197.  1.  Change  lof  to  fourths. 

^  -         4  P  0  In  1  there  are  4  fourths,  and   in   15  there  are  15 

^  ~  ^  times   4   fourths,    or   60    fourths;    60   fourths    +    3 

6J)   4-  .a  —   fit  fourths  =  63   fourths.      Why   do    we  add  3  fourths 

t   -^  t  —    i  ^^  QQ  fourths  ? 

Cliange  to  fractional  form  : 

2.  1|.  9.  9f.  16.  12|i  23.  115 jV 

3.  4f.  10.  3^%.  17.  18^.  24.   aOSeV 

4.  Gf.  11.  514,.  18.  25^V  26.  365/^. 

5.  71.  12.  8.  19.  29^f.  26.  oOOff. 

6.  5f.  13.  7^.  20.  37^V  27.  710|^. 

7.  8|.  14.  6tV  21.   72^.  28.  802.^ 

8.  7^.  15.  10.  22.  90|f  29.  613.25. 

30.  Change  ^^  to  a  mixed  number. 

OQ    .    q  _  02  Since  there  are  3  thirds  in  one,  in  29  thirds  there 

~     ^      are   as  many   ones   as   there   are   3's   in   29,  or  9|  ; 
therefore,  ^3^  _  gj. 

Change  to  integers  or  mixed  numbers  : 

31.  i,  f,  V-  35.  ^^  HK 

32.  ^,  V,  V-         36.  If?,  Y/. 

33.  ¥.  ¥,  \'-         37.   W,  W-. 

34.  5^,  5ji,  ig4.         38.  YJ5.  -7^.. 

Complete  the  following  equations 

43.  38|  =  (     ).       46.  24:j\  =  ( 

44.  1^3  ^  (     ).       47.  i^si  =  ( 

45.  10-[^  ^  (     ).      48.  365i  =  ( 


39.  i|ji   2j^. 

40.  -L|p,  ^ifii. 

41.  5^1-8,    1||1. 

42.  i^s,  ''If''. 

)• 
)• 
)• 

49.  %'  =  (     ). 

50.  lOOlrV  =  (    ). 

51.  4;|^  =  {   )• 

FRACTIONS.  129 

198.  To  change  decimal  fractions  to  common  frac- 
tions. 

1.   (a)  Express  .7  as  a  common  fraction  ;  (b)  change  .125 
to  a  common  fraction. 

/   \      7  _    7  ^®  write  the  figures  of  the  decimal 

^    /  T^'  for  the  numerator,   and   1  with  as 

[o)     .1^0  =  xffij  =  ru  —  ¥•      many  ciphers  after   it  as  there  are 
figures  after  the  decimal  point  for  the  denominator. 

When  desirable,  we  change  to  smallest  denominator,  as  in  (6). 

Change  to  common  fractions  with  smallest  denominators : 


2. 

.25. 

9. 

.39. 

3. 

.35. 

10. 

.43. 

4. 

.85. 

11. 

.38. 

6. 

.75. 

12. 

.50. 

6. 

.24. 

13. 

.95. 

7. 

.72. 

14. 

.05. 

8. 

.60. 

15. 

.03. 

16. 

.375. 

23. 

.0125. 

17. 

.425. 

24. 

.3750. 

18. 

.500. 

25. 

.0875. 

19. 

.625. 

26. 

.1872. 

20. 

.205. 

27. 

.4020. 

21. 

.875. 

28. 

.0075. 

22. 

.945. 

29. 

.15625. 

199.  To  change  common  fractions  to  decimal  frac- 
tions. 

1.  How  many  tenths  in  4  twentieths  ?  In  ^  ?  In  ^  ? 
JJ  =  how  many  lOths  ? 

2.  How  many  tenths  in  ^  ?  In  f  ?  In  f  ?  f  =  how 
many  lOths  ? 

3.  How  many  hundredths  in  ^^^  ?  In  ;jy_  p  Jn  .^^  9 
.^g^  =  how  many  100th s  ? 

4.  How  many  hundredths  in  ^^  ?  In  2V  ?  I^  ^  ?  ^^  A  ? 
^  =  how  many  lOOths  ? 

5.  What  have  you  been  changing  in  these  examples  ?  Since 
lOths  and  lOOths  are  decimal  divisions,  to  what  have  you 
been  changing  the  common  fractions  ? 

200.  To  change  a  common  fraction  to  a  decimal  fraction 
is  to  change  it  to  larger  or  smaller  denominators,  and  to 
express  it  in  the  notation  peculiar  to  decimals. 

Thus,  i  =  ,5,  =  .5  ;  i  =  ,\^  =  .75  ;  -^^^  =  jhj  =  .02. 


130  SCHOOL  ARITHMETIC. 

1.  Change  f  to  a  decimal  fraction,  that  is,  to  a  fraction 
whose  denominator  is  10  or  100  or  1000,  etc. 

1 000  -^8 19^  '^^^  required  denominator  must  be  a 

'  multiple  of  8.     The  least  decimal  denom- 

3  X  125  _  375   _    ^^^        inator  that  is  a  multiple  of  8  is  1000. 
8  X  125        1000        '        *      Hence  |   must   be  changed  to  lOOOths. 

(Art.  189.) 

The  required  denominator  can  be  found  only  by  inspection  or  by  trial. 
Since  it  must  be  divided  by  the  given  denominator,  and  the  given  nu- 
8)3.000     ^"erator  multiplied  by  the  quotient,  we  may,  for  convenience, 
— — —     combine  the  processes  and   obtain  the   required  decimal   at 
.6  lb    Qj^gg^  jjy  dividing  the  numerator  by  the  denominator,  annex- 
ing ciphers,  and  pointing  off  decimal  places  as  in  division  of  decimals. 


(b) 
200)26.00 


2.  Change  -^^  to  a  decimal  fraction. 

In  many  cases,  reducing  a  fraction 
(^)  to  a  smaller  denominator  changes 

^^  =  ^1^=  .13       it  to  a  decimal  fraction,  as  in  (a). 

It  may  also  be  solved  as  in  (b).  •^^* 

Rule. — Annex  ciphers  to  the  numerator  and  divide  hy  the 
denominator,  pointing  off  decimal  places  as  in  division  of 
decifnals. 

Note. — When  the  division  will  not  terminate,  the  remainder  may  be 
expressed  as  a  common  fraction,  or  the  sign  +  may  be  placed  after  the 
decimal  figures  to  show  that  the  division  is  not  complete. 

Thus,  i  =  .33i,  or  .33  +. 


Change  to  decimal  fractions  : 

3.  }.  11.  i\ 

4.  i.  12.  5V 

5.  |.  13.  if, 

6.  i.  14.  U 

7.  |.  15.  H 

8.  f  16.  A 

9.  J.  17.  U' 
10.  ^\.  18.  V, 


19.  tV-  27.  If 

20.  f.  28.  If. 

21.  f  29.  U- 

22.  |.  30.  A- 

23.  j%.  31.  il 

24.  A.  32.  li. 

25.  a.  33.  ^. 

26.  il  34.  Tij. 


FRACTIONS.  131 

201.  To  change  dissimilar  to  similar  fractions. 

1.  Have  I  and  |  the  same  denominators  ?  Have  f  and  |  ? 
Have  f  and  5  ?     Have  f,  J,  and  J  ? 

202.  Fractions  that  have  the  same  denominators  are  said 
to  have  a  common  denominator^  and  are  called  Similar 
Fractions.     They  have  a  common  unit  of  measure. 

203.  Fractions  that  do  not  have  the  same  denominators 
are  called  Dissimilar  Fractions.  Their  units  of  measure 
are  not  the  same. 

Tell  which  are  similar  fractions  : 

1.  h   h   h   h   h   h  3.  I   j\,   f,   i,   t5^,    4. 

2.  h  h  h  f  h  I         4.  tV,  A,  A,  A,  iV  A- 

5.  How  many  sixths  in  1  ?  In  ^  ?  In  ^  ?  Then  what 
common  denominator  may  ^  and  ^  have  ? 

6.  How  many  eighths  in  1  ?  In  ^  ?  In  |  ?  Then  what 
common  denominator  may  |  and  f  have  ? 

7.  When  fractions  have  a  common  denominator,  what  are 
they  called  ? 


Change  to  similar  fractions  : 

8.  i  and  i. 

17.  1  and  f. 

9.  i  and  h 

18.  f  and  f . 

10.  i  and  i. 

19.  t  and  f 

il.  i  and  i. 

20.  t  and  f. 

12.  i  and  i. 

21.  h  h  and  i. 

13.  i  and  f. 

22.  h  h  and  |. 

14.  t  and  f. 

23.  h  f,  and  f. 

16.  1  and  f; 

24.  f,  i,  and  J^. 

16.  1  and  |. 

25.  f,  I  and  /o- 

204.  A  common  denominator  of  two  or  more  fractions  is 
a  Common  Multiple  of  their  denominators. 


132  SCHOOL  ARITHMETIC. 

WRITTEN    EXERCISES. 

205.  1.  Change  f,  f,  and  f  to  similar  fractions. 

2  X  20        40  Since  the   product  of  the  given   denominators 
TT"  ^^7^7         is  a  common  multiple  of  each,  3x4x5,  or  60, 

is  a  common  denominator.     Hence  the  fractions 

3  X  15  __  45         must    be    changed    to   GOths,    which   is  done    by 

4  X  15        60         multiplying  numerator  and  denominator  of  each 

4  X  12        48         ^^  ^^®    quotient    that    arises  from   dividing  the 
^=  —         required    denominator    by    its    given    denoraina- 

5  X  12       60         tor. 

Rules. — 1.  Multiply  numerator  and  denominator  of  each 
fraction  hy  the  number  of  times  its  denominator  is  contained 
in  a  common  multiple  of  all  the  given  denominators.     Or, 

2.  Multiply  numerator  and  denominator  of  each  fraction  hy 
the  product  of  the  denominators  of  all  the  other  fractions. 

Change  to  similar  fractions  : 

2.  h  i,  f. 

3.  i,  I,  A. 

4.  J,  i,  J. 

5.  h  h  f. 
6-  h  i  A- 

7.  f,    .1,   ^. 

8.  f,   i  U- 

9.  tV   -8.  H- 

Ifi      11  7  8  E>  7 

•LO.    "25^      27'     ^¥?     W^y     "6"0- 

206.  Fractions  may  have  more  than  one  common  denomi- 
nator. The  smallest  one  they  can  have  is  called  their  Least 
Common  Denominator.  The  smaller  the  denominator,  the 
greater  the  unit  of  measure. 

207.  The  least  common  multiple  of  the  denominators  of 
several  fractions  is  their  least  common  denominator  (L.  C.  D.). 


10. 

h  h  A. 

11. 

h  h  4. 

12. 

A.  h  1. 

13. 

\h  h  f. 

14. 

i    Ih    .9. 

15. 

i^>  A'  A- 

16. 

^h  .5,  h  U' 

17. 

ii'   ¥Trj   A- 

FRACTIONS.  133 

WRITTEN     EXERCISES. 

208.  1.  Change  f,  f,  and  ^  to  similar  fractious   having 
their  least  common  denominator. 

3  X  18  _  54  We  find  72  to  be  the  L.  C.  M.  of  the  donoini- 

^  ^  jg        1^2  nators,    and   the   least  common  denon)inator 

K  Q        AK  ^^  ^^*^   fractions,    whicli   must,   therefore,   be 

=  —  changed  to  72nds.     This  is  done  by  multiply- 

8x9        75&  iiig     numerator    and    denominator    of    each 

7  X      8  _  56  fraction  by  tlie  number  of  times  its  denomi- 

q  ^       ft^  nator  is  contained  in  the  L.  C.  M. 

B^^  Explain  why  fractions  having  their  least  common  denominator 
have  their  greatest  common  fractional  unit. 

Change  to  similar  fractions  with  their  L.  C  D.  : 

2.  h  i,  i-  8.  i  A,  I.  14.  I  «,  i4,  18. 

3.  h  I  I  9.  H'  -5.  i  18-  l  h  A.  A- 

4.  h  tV.  a-       10-  i  i'  h  f  18-  M.  -8'  H,  iS- 
6.  ^,  S,  i.           IX.  ?,  ^\,  A,  A.         17.  +,  ^,  A,  T^. 

6.  i,  I,  j\.         12.  A.  A.  ih  il       18.  A,  4,  ,'j- 

7.  I,  i,  T^.         13.  f,  f,  I,  -rV-  19.  i  I,  I- 

20.  Change  f  to  a  number  whose  fractional  unit  is  ^. 

21.  How  many  fractional  units  are  there  in  f  if  changed 
to  16ths  ? 

22.  A  fraction  whose  fractional  unit  is  an  eighth  has  the 
numerator  6.     What  is  the  value  of  the  fraction  in  fourths  ? 

23.  What   fraction   has  sixteen   fractional   units,  each  of 
which  is  ^^  ? 

ADDITION. 

209.  1.  Ella  has  If,  and  Jane  has  $f .     How  much  have 
they  both  ? 

2.  Mary  paid  ||  for  a  knife  and  $|  for  a  book.     How  much 
did  she  pay  for  both  ? 

3.  George  bought  f  of  an  acre  from  one  man,  and  f  of  an 
acre  from  another.     How  much  land  did  he  buy  ? 


134:  SCHOOL  ARITHMETIC. 

4.  "When  fractions  have  different  denominators,  what  must 
be  done  to  them  before  they  can  be  added  ?     Why  ? 

6.  Can  :|^  of  a  dollar  and  -J  of  a  bushel  be  added  ?  Why 
not? 

210.  Principle. — Fractions  can  he  added  only  when  they 
express  like  quantities,  and  have  a  common  denominator. 

WRITTEN     EXERCISES. 

211.  1.  Find  the  sum  of  |,  f,  and  \\. 

^      *  ^  Changing  the  fractions  to  similar  ones  hay- 

's     4  b  ing  their  least  common  denominator,  we  have 

i^— ff  24,  85,  and  22  fortieths,  and  the  sum  of  these 

2.  What  is  the  sum  of  3^,  7^,  and  12f  ? 

^          1^  Changing  the  fractions  to    similar    ones 

•1)        •  iV  having  their  least  common  denominator,  we 

13f  =  12|f  find  their  sum  to  be  f|,  or  IH;  the  sum  of 

yoj^i  the  integers  is  22 ;  and  the  sum  of  both  is  23|i , 

Supply  the  words  that  are  wanting  in  the  following 

Rule. — Change  the  given  fractions  to ,  add 

their ,  and  write   the  sum   over  the  . 

When  there  are  integers  or  mixed  numbers,  add and 

separately,  and  then the  two  sums. 

Add  the  following: 

3.  h  h  i,  i-  12-  i\'  H.   ih  -35 

4.  I,  f,   tV.   «• 
8.  h  .7,  A,  A- 
6-  f  A.  i,  T%- 

8.  I,  i,  J^,  «. 

9-  A.  \h  ih  A- 
10.  i  A,  ih  A- 
U-  h  i  A.  i- 


13. 

n, 

4|,    2t. 

14. 

61, 

5i,    3. 

15. 

8i 

Q-A,    2.5. 

16. 

3i 

8.7,    121. 

17. 

131 

,    17tV    ^0,5^. 

18. 

221. 

.    181,    7t\. 

19. 

30f 

,    20^,    40.3. 

^0, 

5t, 

2A,    8Jf. 

FRACTIONS.  135 

21.  One  week  Jasper  earned  |;3|,  Homer,  $4.7,  David 
$12|,  and  James,  $0^^.     How  much  did  all  earn  ? 

22.  Mary  weighs  75^  pounds,  Edna  12f  pounds  more  than 
Mary,  and  Charles  as  much  as  both.  How  much  does 
Charles  weigh  ? 

23.  The  product  of  two  numbers  is  the  sum  of  258^  and 
173|,  and  one  of  the  numbers  is  24.     Find  the  other. 

24.  Find  the  distance  around  a  rectangular  field  whose 
length  is  281f  rods,  and  whose  width  is  190f  rods. 

25.  A  certain  minuend  would  be  iJ,  if  it  were  ^  less. 
What  would  it  be  if  increased  by  ^  ? 

26.  A  locomotive  runs  354f  miles  every  other  day.  How 
far  does  it  run  in  a  week,  not  counting  Sunday  ? 

27.  What  is  the  minuend  when  the  remainder  is  17^,  and 
the  subtrahend  10^^  ? 

28.  The  number  of  fractional  units  in  the  greater  of  two 
fractions  is  15,  the  number  in  the  less  is  G.  The  denomi- 
nator of  the  greater  fraction  is  27,  and  that  of  the  other  is  18. 
Find  the  sum  of  the  fractions. 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

212.  1.  Mr.  A  has  5  fields,  the  smallest  of  which  contains 
6f  acres,  and  the  largest  12.375  acres.  Each  of  the  others 
contains  9^  acres.     How  much  land  has  Mr.  A  ? 

2.  John  has  $3j'^,  Harry  has  $li  more  than  John,  Ben 
has  13.05  more  than  both,  and  their  father  has  $10.25  more 
than  the  three  boys  together  have.  How  much  have  all 
four  ? 

3.  A  man  earned  $16.66f  in  January.  Each  month  there- 
after he  earned  $16,125  more  than  he  earned  the  preceding 
month.     How  much  did  he  earn  in  a  year  ? 

4.  A  lady  spent  ^  of  her  money  for  shoes,  f  for  a  dress, 
and  f  for  a  hat.  If  she  spent  $37.30  for  all,  how  much  had 
she  left  ? 

5.  Find  the  value  of  3  x  (6f  +  5.2)  -  7.5  ^  (.5  +  1). 


136  SCHOOL  ARITHMETIC. 

6.  'A  can  do  |  of  a  piece  of  work  in  a  clay,  B  |  of  it,  C  J  of 
it,  and  D  ^  of  it.  In  what  time  can  they  do  the  work,  all 
working  together  ? 

7.  I  sold  to  different  customers  13f  gallons,  11.5  gallons, 
15^  gallons,  3^  gallons,  and  11.75  gallons  of  oil.  How  many 
gallons  did  I  sell  to  all  ? 

8.  Find  the  sum  of  all  the  proper  fractions  that  can  be 
formed  having  one  figure  each  in  the  numerator  and  denom- 
inator. 

9.  Ten  is  added  to  a  certain  mixed  decimal.  The  point  is 
then  moved  one  place  to  the  left  and  10  is  added.  The  sum 
is  equal  to  4.5  times  the  original  number.  Find  the  original 
number. 

SUBTRACTION. 

213.  1.  Ella  has  $f,  and  Jane  has  $f.  How  much  more 
has  Ella  than  Jane  ? 

2.  Mary  paid  $^  for  a  knife  and  $f  for  a  book.  How  much 
did  the  book  cost  more  than  the  knife  ? 

3.  A  man  who  had  |  of  an  acre  of  land  sold  f  of  an  acre. 
How  much  had  he  left  ? 

4.  When  fractions  have  different  denominators,  what  must 
be  done  to  them  before  they  can  be  subtracted  ?     Why  ? 

5.  Can  f  of  a  dollar  be  subtracted  from  f  of  a  foot  ?  A¥hy 
not  ? 

214.  Principle. — Fractions  can  he  subtracted  only  tvhen 
they  express  like  quantities,  and  have  a  common  denomi- 
nator. 

WRITTEN    EXERCISES. 

215.  1.  Subtract  f  from  |. 

1  =  11 

3  2Jl  Changing  to  similar  fractions,  we  have  32 

„  thirty-sixths,    and    27    thirty-sixths,    whose 

■^  difference  is  3-5. 


FRACTIONS.  137 

2.  From  6|  take  4|. 

(a)         (b)  The  fractions  must  be  made  similar,  as  in  (a). 

g3  _.  go    :::^  533.      ^^  caniiot  be  taken  from  9^4,  hence  1,  or  ||,  is  taken 

42  _  41.6.  =3  4'R      ^*'0"^   ^  **"*^  added   to  /<,  making   \\,  as   in  (b). 

^  ~     2^^  ""     ^^     Then  i2  ^''""i  i5  leaves  i^.  tt"^  ^  from  5  leaves  1. 

l^J     Hence  the  remainder  is  UJ, 

1^  In  practice,  the  numbers  under  (b)  should  not  be  written;  the 

work  should  be  done  mentally. 

Supply  the  blanks  in  the  following 

Rule. — Change  the  fractions  to ;  fi7id  the 

difference  between  the ,  and  write  it  over  the . 

Wheii  there  are  mixed  numbers  or  integers,  subtract  frac- 
tions and  integers  separately. 

Query. — May  integers  or  mixed  numbers  be  changed  to  fractional 
form  and  subtracted  according  to  the  first  part  of  the  rule  ? 

Find  the  value  of : 


3.  f- 

f. 

10. 

l+i-h 

17.  G.7  -  ^. 

4.  A  ■ 

-A- 

11. 

.9  +  3-f 

18.  7f  -  0|. 

8.  -,V  - 

-sV 

12. 

f  -  A  +  4- 

19.  21.7  -  16jV 

6.  A  - 

-i- 

13. 

A-i--i- 

20.  14i  -  11. 

7.  H- 

-A- 

14. 

H  +  1  -1- 

21.  12J  -  3ft. 

8.  5- 

■n- 

15. 

"i  -  n- 

22.  2i  +  3^  +  5^. 

9-M- 

-i-v 

16. 

13j^  -  4.26. 

23.  4i  +  6f  -  3.7. 

24.  Ella  is  lOf  years  old,  and  Nellie  is  19|  years  old. 
How  much  oUler  is  Nellie  than  Ella  ? 

25.  A.  pole  14|  feet  long  was  broken  into  two  pieces,  one 
of  which  was  5|  feet  long.     How  long  was  the  other  ? 

26.  A  train  ran  from  Lynchburg  to  Roanoke  in  1|  hours; 
it  reached  Roanoke  at  9  a.m.     At  what  time  did  it  start  ? 

27.  Carl  bought  a  book  for  $2yV,  a  knife  for  $|,  a  hat  for 
$1|,  and  a  coat  for  $2^.  If  he  had  a  ten-dollar  bill  at  first, 
how  much  had  he  left  ? 

28.  The  sum  of  two  numbers  is  21f,  and  one  of  them  is 
8f .     What  is  the  other  ? 

29.  What  fraction  added  to  ^  +  |  +  f  will  make  2^^  ? 


138  SCHOOL  ARITHMETIC. 

30.  The  minuend  is  J|,  and  tlie  remainder  is  |.  What  is 
the  subtrahend  ? 

31.  Two  fractions  have  the  common  denominator  12.  One 
has  8  fractional  units,  the  other  11.  How  much  greater  is 
one  than  the  other  ? 

32.  Subtract  ^  from  the  greatest  possible  fractional  unit. 

33.  If  4  is  added  to  numerator  and  denominator  of  f,  how 
much  is  the  value  of  the  fraction  increased  or  diminished  ? 
How  much  if  the  fraction  is  |  ? 

SUPPLEMENTARY    EXERCISES      (FOR    ADVANCED    CLASSES). 

216.  1.  If  numerator  and  denominator  of  |  are  each 
diminished  by  2,  will  the  value  of  the  fraction  be  increased 
or  diminished,  and  how  much  ? 

2.  What  number  is  that,  f  of  which  exceeds  ^  of  it  by  llf  ? 

3.  A  owns  f  of  a  store,  and  B  the  remainder.  If  f  of  the 
store  is  worth  $575  more  than  .5  of  it,  what  is  the  value  of 
B's  share. 

4.  What  number  is  that,  to  which  if  its  f  and  its  .25  are 
added,  the  sum  wiirbe  170  ? 

5.  The  difference  between  the  subtrahend  and  minuend  is 
5|-.  If  the  subtrahend  is  8^^^  less  than  9y\,  what  is  the  min- 
uend ? 

6.  AVhat  fraction  is  as  much  larger  than  f,  as  f  is  less 
than  I  ? 

7.  The  highest  score  in  an  inning  was  f  of  the  total,  and 
the  next  highest  was  ^^-g^  of  the  total  less.  The  scores  in 
these  two  innings  differed  by  5  runs.  What  was  the  total 
score  ? 

MULTIPLICATION. 

217.  1.  James  has  $^,  and  Henry  has  twice  as  much. 
How  much  has  Henry  ?     Then  2  times  -^  =  {     ). 

2.  Rose  has  ^  of  a  pie,  and  Orville  has  three  times  as  much, 
How  much  has  he  ?    Then  3  x  I  =  {     ), 


FRACTIONS.  139 

3.  May  worked  f  of  a  week,  and  Arthur  worked  4  times  as 
long.     How  long  did  he  work  ? 

4.  How  much  is  5  times  3  twentieths  ? 

6.  5  X  j^  =  J^.  Have  ^  and  ^f  the  same  denominators  ? 
How  do  their  numerators  compare  ?  How  do  the  fractions 
compare  in  value  ?  Which  has  the  greater  number  of  frac- 
tional units  ? 

218.  Pkinciples. — 1.  Multiplying  the  numerator  of  a 
fraction  hy  any  number  multiplies  the  value  of  the  fraction 
hy  that  number . 

For,  since  the  numerator  tells  how  many  fractional  units  are  taken, 
multiplying  it  multiplies  the  number  of  fractional  units,  each  of  which 
has  the  same  size  or  value  as  before. 

Conversely, 

2.  Dividing  the  numerator  of  a  fraction  hy  any  number 
divides  the  value  of  the  fraction  by  that  number. 

For,  since  the  numerator  tells  how  many  parts  are  taken,  dividing  it 
divides  the  number  of  parts,  each  of  which  is  of  the  same  size  as  before. 

WRITTEN     EXERCISES. 

219.  1.  Multiply  ^  by  9. 

q  1.—  27—  17         ^  times  3  twentieths  =  27  twentieths,  just 

»   X  2-0  -  tl5-  -  -LjV    j^g  9  tjjjjgg  3  tops  =  27  tops. 

2.  Multiply  3i  by  7. 

3|-  When  the  multiplier  is  an  integer,  it  is  often  more 

_7_  convenient   to   multiply   the    integer    and  fraction 

21     =7x3  separately,  and  then  add  the  products.     Or  we  may 

1|^  =  7   X  :^  proceed  as  in  the   first  example,  thus :    7  x  3^  = 

m  =  1  x3i  7x^.^  =  ^.^  =  221. 

Multiply  the  following : 

3.  I  by  12.  8.  ^^  by  15.  13.  5f  by  10. 

4.  I  by  13.  9.  \l  by  11.  14.  llf  by  14. 

5.  1^  by  8.  10.  t\  by  21.  15.  13|  by  5. 

6.  y«3  by  11.  11.  II  by  38.  16.  15|  by  7. 

7.  f^  by  16.  12.  2J  by  8.  17.  23f  by  12. 


140  SCHOOL  ARITHMETIC. 

18.  Multiply  eacli  of  the  following  by  its  own  denomi- 
nator, and  note  the  results  :  |,   f,   f,   -g,    f,   ^,    j,   -^g. 

19.  How  does  it  affect  a  fraction  to  multiply  it  by  its  de- 
nominator ?     Find  by  trial. 

20.  Multiply  eacli  of  the  following  mixed  numbers  by  the 
denominator  of  the  fractional  part,  and  compare  each  mul- 
tiplicand with  its  product :  2^,   1:^,   4^,   3|,   6f. 

21.  How  does  it  affect  a  mixed  number  to  multiply  it  by 
the  denominator  of  the  fractional  part  ? 

Note. — As  regards  their  form,  it  is  usual  to  make  a  distinction  be- 
tween simple  and  complex  fractions.  They  are  called  simple  when 
numerator  and  denominator  are  integers  ;  otherwise  complex. 


22.  Multiply  the  complex  fraction  —  by  4. 

0 

4  X  I  ^   V\^  3 
5  5  5* 

23.  Change  -^  to  a  simple  fraction. 

2    ^   Q  6  2  1  A  fraction  is  made  integral  by  multi- 

j o  =  "T^  ^^  T^  ~  fi^     plying    it    by    its    denominator.      Hence 

we  multiply  both  numerator  and  denomi- 
nator by  the  denominator  of  the  fraction  §;  that  is,  by  the  denominator 
which  is  in  the  numerator  of  the  complex  fraction. 


Change  to 

simple 

fractions  ; 

-!■ 

28. 

i 
8* 

"t 

29. 

15* 

26.  |-. 

30. 

5* 

«.i. 

31. 

H 

4' 

32. 
33. 
34. 
35. 


5| 

7' 

±0. 

1 1 

20' 

17| 

22' 

20^ 

.21  * 


FRACTIONS.  Ul 


36.  Multiply 

12 

hyl 

To  multiply  12  by  |  is  to  find  |  of  12,  or  2 

12  -^  3  =  4 

times  ^  of  12. 

2x4=8 

^  of  12  =  12  -h  3,  or 

4  ;    and    J  of   12  =  2 

or 

times  4,  or  8. 

12  x  J  =  V  = 

:   8 

Or,  since  f  x  12  =  12 
J,  or  i/,  or  8. 

X  f,  we  have  12  times 

Find  the  vali 

Lie 

of: 

37.  15  X  |. 

44.  tV  X  23. 

61.  7f  X  30. 

38.  16  X  f. 

46.  -^  X  17. 

62.  of  X  42. 

39.  12  X  f. 

46.  2i  X  16. 

63.  9.8  X  14. 

40.  14  X  f 

47.  18  X  3i. 

64.  18f  X  12. 

41.  18  X  f. 

48.  21  X  6f. 

66.  '^  X  16. 

42.  24  X  f. 

49.  24  X  6f. 

66.  25  X  y. 

0 

43.  27  X  4. 

60.  25  X  ^. 

67.  42  X  4. 

68.  Multiply  1  by  4. 

3  X  4  _  V-  _ 

12 

Multiplying  the  numerator  of  a  fraction  by 
!      anv  number   multinlies   the   fraction   bv   that 

5  5  35     number.     (See  Art.  218.)    Hence  we  multiply 

3  by  ^  (as  in  example  36),  which  gives  \\  Writing  this  product  over 
the  denominator,  this  is  readily  changed  to  the  simple  fraction  ^f .  (See 
example  23.) 

Multiply  the  following  : 

59.  4  by  h  63.  jV  by  f        67.  6  by  f. 

60.  I  by  i.        64.  f^  by  f.        68.  12  by  |. 

61.  I  by  i        65.  2i  by  H.       69.  15^  by  I 

62.  I  by  f .  66.  3f  by  2^^.  70.  f  by  2^. 
(U^*  In  the  product  of  ?  x  f ,  or  in  the  product  of  any  two  or  more 

fractions,  it  will  be  seen  that  the  numerator  of  the  product  is  the  prod- 
uct of  the  numerators  of  the  factors,  and  the  denominator  the  product 
of  the  denominators  of  the  factors. 

Hence  the  following  convenient  method  may  be  used  : 

'RvLE.^Change  all  integers  and  mixed  numbers  to  frac- 


142  SCHOOL  ARITHMETIC. 

tional  form ;  then  write  the  product  of  the  numerators  over 
the  product  of  the  denominators. 

Notes. — 1.  The  work  may  often  be  very  much  shortened  by  cancella- 
tion. 

2,  The  sign  x  after  a  fraction  is  sometimes  read  "of."  Thus,  |  x 
$3  may  be  read  "four  fifths  of  three  dollars."  In  expressions  like  ^  of 
^  the  sign  x  may  be  substituted  for  "of." 

Find  the  value  of  : 


71. 

f  xf 

76. 

f  x|. 

81.  S^a^  X  34. 

72. 

4xf 

77. 

u 

xif. 

82.  lOj^  X  33. 

73. 

f  xf 

78. 

if 

xU- 

83.  21f  X  6|. 

74. 

tV  X  ^^. 

79. 

H 

XtV 

84.  13t  X  IVj-. 

75. 

A  X  A. 

80. 

.9 

xA. 

85.  324  X  ^. 

86.  if  X 

:f  X  Y. 

91. 

f  X  4^  X  i|. 

87.  if  X 

:  U 

xf. 

92. 

3i  X  2^  X  ^. 

88.  41  X 

n 

xi 

93. 

f  X  3.1  X  5i 

89.  1  X 

|x 

3^. 

94. 

90  X  3i  X  If 

90.  f  X 

1  X 

n- 

95. 

if  X  f  X  4^. 

96.  Find  the  weight  of  2f  bushels  of  oats,  allowing  32 
pounds  to  a  bushel. 

97.  If  a  man  can  walk  3 A  miles  in  an  hour,  how  far  can 
he  walk  in  7f  hours  ? 

98.  A  and  B  paid  1160  for  a  horse.  If  A  paid  /„  of  the 
cost,  how  much  money  did  B  pay  ? 

99.  The  multiplier  is  -J,  and  the  multiplicand  is  -^.  AVhat 
is  the  product  ? 

100.  Mr.  E  owned  f  of  a  mill  which  w^as  valued  at  $12000. 
He  sold  B  y\  of  his  share.  What  was  the  value  of  the  part 
retained  ? 

101.  A  farmer  bought  12f  acres  of  land  at  $37^  an  acre, 
and  sold  it  at  $52^  an  acre.     How  much  did  he  gain  ? 

102.  A  man  who  had  f  of  an  acre  of  land  sold  |  of  his 
share  at  the  rate  of  $300  an  acre.   How  much  did  he  get  for  it  ? 


FRACTIONS.  143 

103.  What  part  of  a  gallon  is  f  of  f  of  a  gallon  ? 

104.  If  a  ton  of  hay  is  worth  |15f,  what  is  the  value  of 
7.5  tons  ? 

105.  When  cloth  is  worth  $1^  a  yard,  how  much   must  he 
paid  for  |  of  a  yard  ? 

106.  A  has  f  as  much  money  as  B,  who  has  §  of  $80.     How 
much  money  has  A  ? 

107.  Find  the  cost  of  12^  pounds  of  butter  at  18f  cents  a 
pound. 

$ .  18f  In  examples  like  this  it  is  often  better  to  multiply  as  in  the 

13i^  solution  given.     Thus, 

2iQ  12  X  18  cents  =  216  cents. 

9  I    X  18  cents  =  9  cents. 

9  12  X  f  of  a  cent  =  9  cents. 

|.  i    X  i  of  a  cent  =  |  of  a  cent. 
The  sum  of  all  is  234|  cents,  or  $2.34|. 


$2.34f 

108.  What  is  the  value  of  7  barrels  of  sugar,  each  con- 
taining 344^  pounds,  at  |.04f  a  pound  ? 

109.  What  is  the  price  of  a  1000-mile  book  of  tickets  at 
If  cents  a  mile  ? 

110.  If  a  cubic  foot  of  water  weighs  62|  pounds,  and  iron 
is  7^Q  times  as  heavy  as  water,  what  is  the  weight  of  a  cubic 
foot  of  iron  ? 

111.  How  many  pounds  of  ship-biscuit  will  be  required 
for  a  cruise  of  30  days,  if  the  daily  allowance  is  j\  of  a  pound 
to  each  of  the  250  men  in  the  crew  ? 

112.  If  the  freight  rate  is  If^  cents  a  ton  for  each  mile, 
what  will  it  cost  to  ship  5^  tons  of  produce  100  miles  ? 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

220.  1.  A  ship  sailed  3  days,  each  day  sailing  l^j  times 
as  far  as  on  the  preceding  day.  If  she  traveled  55  miles  the 
first  day,  how  far  did  she  travel  the  last  two  days  ? 

2.  How  much  must  be  paid  for  the  rent  of  a  store  for  2f 
years  at  $62^  a  month  ?  . 


144  SCHOOL  ARITHMETIC. 

3.  If  a  man  plants  J  of  an  acre  of  corn  in  a  day,  how  many 
acres  will  7  men  plant  in  3.5  days  ? 

4.  I  bought  51  pounds  of  sugar  at  6|  cents  a  pound,  and  | 
as  much  coffee  at  4  times  the  price  paid  for  the  sugar.  What 
did  I  pay  for  the  coffee  ? 

5.  Divide  $4.09  between  two  boys  so  that  one  will  receive 
40  cents  more  than  twice  what  the  other  receives. 

6.  If  I  breathe  17  times  a  minute,  and  take  in  at  each 
breath  f  of  a  quart  of  air,  how  many  quarts  of  air  do  I  need 
in  ^  hour  ? 

7.  If  each  soldier  walks  |  of  a  mile  in  ^  of  an  hour,  what  is 
the  combined  distance  marched  by  a  regiment  of  800  soldiers 
in  the  same  time  ? 

8.  The  factors  of  the  multiplicand  are  2,  3,  4,  5,  and  f ; 
those  of  the  product  are  f,  and  the  prime  factors  of  144. 
What  is  the  multiplier  ? 

9.  A  clock  loses  lj\  minutes  in  a  day.  If  it  is  correct  at 
noon  on  the  4th  of 'July,  what  time  will  the  clock  indicate 
at  noon  on  the  14th  of  July  ? 

10.  A  clock  loses  J  of  a  second  every  5  minutes.  How 
much  will  it  lose  in  5  days  ^nd  12  hours  ? 

11.  Prove  that  the  sum  of  two  fractions,  the  numerator  of 
each  of  which  is  1,  is  equal  to  the  siwi  of  the  denominators 
divided  by  their  product. 

Divisioisr. 

221.  1.  A  man  having  4  fifths  of  an  acre  divided  it  into  2 
equal   lots.       How   much   land    was   in   each    lot  ?       Then 

*-3  =  (     )■ 

2.  A  lady  divided  3  fourths  of  a  pie  equally  among  3  boys. 
How  much  did  each  boy  get  ?     Then  ^  ~  d  =  (     ). 

3.  Mrs.  A  gave  |  of  a  cake  to  4  girls.  How  much  did 
each  get  ?    Then  «  ^  4  =  (     ). 

4.  -|  -V-  4  =  -f.     Are  the  fractional  units  of  this  dividend 


FRACTIONS.  145 

and  quotient  equal  in  size  ?  Are  they  equal  in  number  ? 
The  number  in  the  quotient  is  what  part  of  the  number  in 
the  dividend  ?  Then,  how  is  a  fraction  divided  by  an  inte- 
ger ?     (See  Art.  218.) 

WRITTEN    EXERCISES. 

222.  1.  Divide  j*  ^7  3. 

Dividing  the  numerator  of  a  fraction  by  any  number 
2V  "^  ^  ~  ^TF     divides  the  value  of  the   fraction   by   that  number. 
(Art.  218.) 

2.  Divide  f  by  6. 

Dividing  the  numerator  by  6,  a.s  in  the  pre- 

„    .    r,  f_ 3__  jL     ceding  example,  we  get  a  complex  fraction  as 

**  8  a  quotient,  which  is  changed  to  a  simple  frac- 

tion, as  in  Art.  219. 
Find  the  quotients  of  : 

II  -  15.  17.  «^  -  35. 

U  -  18.  18.  4  -  12. 

U  -  14.  19.  t  -  9. 

A-f-4.  20.  ill -24. 

if  -  5.  21.  If  §  -  28. 

il-9.  22.  11^^32. 

i  -  11.  23.  ei  -  36. 

24.  Find  the  quotient  of  127f  divided  by  9. 

The  dividend  may  be  changed  to  fractional  form 
9)127f  a,nd  divided  as  in  the  preceding  examples.     But  it 

JT  is  often  more  convenient  to  proceed  as  follows : 

14—  =  14^T         9  is  contained  in  127  fourteen  times,  with  a  re- 
mainder of  1^,  and  1^  -i-9  =  ^-.    Hence  the  quotient 
is  UjV 
Divide  the  following: 

25.  32i  by  4.  31.  134^  by  12.         37.  ^£  by  5. 

26.  48|  by  3.      32.  176f  by  5.     38.  2f  by  19. 

27.  65i  by  5.      33.  248^  by  6.     39.  7|  by  18. 

28.  73|  by  6.      34.  325^  by  3.     40.  if  by  5. 

29.  92|  by  7.     35.  540|  by  7.     41.  170f  by  20. 

30.  93f  by  9.     36.  809^^  by  8.    42.  2001-  by  10. 

10 


3.  1  -^  2. 

10. 

4.  H-5. 

11. 

5.  if -4. 

12. 

6.  li-7. 

13. 

7.  f  i  -^  9. 

14. 

8.  IS -14. 

15. 

9.  U  -  13. 

16. 

146  SCHOOL   ARITHMETIC. 

43.  A  man  earned  $13594^  in  9  years.     What  was  his  aver- 
age yearly  income  ? 

44.  How  often  is  f  contained  in  9  ? 

9  r=  ^  Changing  9  to  fourths,   we    have  36    fourths  -t-  3 

sfi  _i_  3  _  22     fourths.     36  fourths  contains  3  fourths  12  times,  just  as 
$36  contains  $3  twelve  times. 

45.  How  often  is  -fj  contained  in  -|  ? 

■9  =  f^  Changing  dividend  and  divisor  to  similar  fractions, 

^  =  ^  we  proceed  as  in  example  44.    Any  example  in  divi- 

88  _i_  1  8  -_  ^     sion  of  fractions  may  be  solved  in  this  manner. 

Find  the  quotients  of  : 


46.  i^i. 

52.  a  -^  |. 

58.  0-i--^. 

47.  i  -i-  i. 

53.  5  -^  f . 

59.  3f  -  3i. 

48.  t  -^  f . 

54.  8  -  f 

60.  4f  -^  oi 

49.  A  -i-  i. 

55.  2i  -=-  i. 

61.  9i  -  IJ. 

50.  1  -  f 

56.  7i  -^  |. 

62.  7|  H-  Jf 

51.  T^  ^  f 

57.  3f  -  |. 

63.  ^\  -^  2i. 

64.  How  many  times  is  f  contained  in  |  ? 

Since  i  is  contained   in   1  eight 
^  "^  E^  ^^  "  times,  I  is  contained  in  1  one-third  of 

1  -7-  f  =  1^    (divisor  inverted)         8  times,  or  f  times  ;  and  in  i  of  1, 
|^-f-^=:4  X  1=  |-|,  or  2y\      o^'  o>  it  is  contained  f  of  |  times,  or 

'li  times. 
The  inverted  divisor  shows  how  many  times  the  divisor  is 
contained  in  1 ;  it  is  called  the  reciprocal  of  the  divisor. 
Rule. — Multiply  the  inverted  divisor  hy  the  dividend. 

Mixed  numbers  and  integers  must  be  expressed  in  fractional  form. 
Cancellation  should  be  used  whenever  possible. 

Find  the  value  of  : 


65.  rV  - 

-tV 

72.  Jt  ^  M. 

79.  51}  -^  7f. 

66.  A  - 

-A. 

73.  ?|  -^  |}. 

80.  A  -  12. 

67.  A- 

-^■ 

74.  7  -  A- 

81.  Jf  H-  20. 

68.  J} - 

-i|. 

75.  8  -  A. 

82.  1^  -J-  60. 

69.  H  - 

■r{h 

76.  3f  --  |. 

83.   15|  -r  7. 

70.  M  - 

-if. 

77.  8|  -  i 

84.  181^  -V-  2.2. 

71.11- 

-»• 

78.  Uf  ^  A. 

85.  7.5-^  22 J.  - 

FRACTIONS.  147 

86.  Find  the  value  of  -^. 

I 
This  expression  is  equivalent  to  2^  -f-  3,  and  may  be  treated  accord- 
ingly.    (Note  3,  Art.  185.) 

87.  What  is  the  quotient  of  ^8^  -  f  ? 

■y\  -4-  1^  =  f  This  process  is  the  converse  of  that  employed  in 

multiplication  of  fractions,  second  rule. 

The  dividend  -,*5  is  the  product  of  two  factors,  one  of  which  is  §. 
Since  the  numerator  8  is  the  product  of  two  factors,  one  of  which  is  2, 
the  other  factor  is  8  -j-  2,  or  4,  which  is  the  numerator  of  the  required 
quotient.  Since  the  denominator  15  is  the  product  of  two  factors 
(or  denominators),  one  of  which  is  3,  the  other  factor  is  15  -i-  3,  or  5, 
which  is  the  denominator  of  the  required  quotient.  Hence  the 
quotient  is  ^. 

^°  This  is  the  most  convenient  method  when  the  numerator  and  de- 
nominator of  the  dividend  are  respectively  multiples  of  the  numerator 
and  denominator  of  the  divisor. 

Divide,  using  this  method  : 

88.  A  by  f .  92.  U  by  A-  »6.  ^t\  by  2|. 

89.  It  by  |.  93.  M  by  A-  97.  7^  by  1^. 

90.  if  by  A.  94.  Hbyf  98.  U^yU- 

91.  if  by  f .  95.  3A  by  |.  99.  «§  by  if 

100.  At  $2f  a  day,  how  long  will  it  take  to  earn  $37^  ? 

101.  If  Thomas  can  walk  3^  miles  au  hour,  in  what  time 
can  he  walk  20  miles  ? 

102.  A  man  gave  |.5G  for  sheep,  paying  $5J  a  head.  How 
many  did  he  buy  ? 

103.  If  a  bankrupt's  property  is  worth  $3100,  and  his 
debts  amount  to  $7000,  how  many  cents  on  a  dollar  can 
he  pay  ? 

104.  A  man  paid  $55  for  3f  tons  of  hay.  What  was  the 
price  a  ton  ? 

105.  A  horse  traveled  24J  miles  in  4  hours.  At  what  rate 
an  hour  was  that  ? 


106.  How  many  times  must  f  be  added  to  itself  to  make  7^? 

107.  How  many  times  can  f  be  subtracted  from  7  ? 


148  SCHOOL  ARITHMETIC. 

108.  A  man  having  10  acres  of  land  sold  each  of  several 
men  f  of  an  acre,  and  had  2^  acres  left.  To  how  many  men 
did  he  sell  ? 

109.  The  distance  around  a  barn  is  213|^  feet.  How  wide 
is  the  barn,  if  it  is  60}  feet  long  ? 

110.  I  bought  25  bushels  of  i^otatoes  at  $f  a  bushel,  and 
sold  them  for  $18|.     How  much  did  I  gain  on  one  bushel  ? 

111.  What  must  3|  be  multiplied  by  to  make  11^  ? 

112.  A  man  bought  396  pounds  of  oats,  at  $f  a  bushel. 
Allowing  32  pounds  to  a  bushel,  what  did  he  pay  for  them  ? 

113.  A  man  paid  $1359f  for  21f  acres  of  land.  How 
much  was  that  an  acre  ? 

114.  The  divisor  is  .8,  the  dividend  -f.     Find  the  quotient. 
116.  When  eggs  are  worth  18J  cents  a  dozen,  how  much 

must  be  paid  for  5  dozen  and  6  eggs  ? 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

223.  1.  A  boy  bought  lemons  at  the  rate  of  8  for  7  cents, 
and  sold  them  at  the  rate  of  7  for  8  cents.  If  he  made  $1.35 
in  7  days,  how  many  did  he  sell  each  day  ? 

2.  A  merchant  purchased  a  cargo  of  flour  for  $2173^,  and 
sold  it  for  If  of  its  cost,  thereby  losing  $^  on  a  barrel.  How 
many  barrels  did  he  purchase  ? 

3.  The  product  of  a  number  multijolied  by  3  is  how  many 
times  the  product  of  the  same  number  multiplied  by  f  ? 

4.  A  owns  2^  times  as  much  land  as  B,  C  owns  1^  times  as 
much  as  both  A  and  B,  and  D  owns  3f  times  as  much  as  B 
and  C.     If  B  has  36^  acres,  how  much  has  J)  ? 

5.  A  boy  bought  |^  of  a  bushel  of  nuts,  and  sold  -|  of  them  for 
what  he  paid  for  all,  and  the  remainder  at  cost.  If  he  gained 
$1|  by  the  transaction,  how  much  money  had  he  invested  ? 

6.  Change  jf-^  to  a  decimal,  ^nd  divide  by  5000. 

7.  The  owner  of  ^j-  of  a  mine  sold  -^q  of  his  share  for 
$40,500.  What  should  he  who  owns  f  of  the  mine  get  for 
I  of  his  share  ? 


FRACTIONS.  149 

8.  A  can  mow  a  field  in  3  days,  B  in  4  days,  C  in  5  days, 
and  D  in  6  days.  If  A  can  earn  $20  a  week,  how  much  can 
B,  C,  and  D  together  earn  in  the  same  time  ? 

9.  If  8f  tons  of  hay  are  worth  36  sheep,  and  11  sheep  are 
worth  2  cows,  and  9  cows  are  worth  1200,  how  many  dollars 
is  hay  worth  a  ton  ? 

10.i  +  L  +  i:-^^^_  =  (    ). 
5  1  +  1.5 

11.  (a)  .00001  -^  10000  =  (  ) ;  (b)  1000  -^  .0001  =  (  ) ; 
(c)  .001  -7-  .000001  =  (     )  ;  (d)  400  -^  10000  =  (     ). 

12.  Find  value  of  2f  -^  f  x  24^\  x  0. 

13.  The  sum  of  two  numbers  is  1000  ;  the  difference  is 
648.     Divide  twice  the  larger  by  J. 

14.  A  owns  f  of  a  farm,  and  B  the  remainder,  f  of  the 
difference  between  their  shares  is  $10500.  What  is  the  value 
of  the  farm  ? 

15.  The  minuend  is  8  times  ^  x  .6  ;  the  remainder  is  i  of 
.7  -^  -J.     What  is  the  subtrahend  ? 

16.  A  can  dig  16|-  rows  of  potatoes  in  a  day,  and  B  can  pick 
33 J  rows  in  a  day.  If  A  has  93.5  rows  dug  when  B  begins, 
how  many  rows  must  B  pick  before  he  overtakes  A  ? 


ILLUSTKATIVE     SOLUTIONS, 

With  Problems  for  Practice. 


o« 


524.  The  solution  of  all  problems  in  the  applications  of 
arithmetic  requires  Analysis  of  some  kind.  A  type  of  aritJi- 
metical  analysis  much  employed  involves  the  process  of 
reasoning  from  a  given  number  to  one,  and  then  passing  from 
07ie  to  the  required  numher. 

225.  As  many  formal  solutions  begin  with  a  '' since  ^'  and 
end  with  a  "therefore/'  it  is  convenient  to  represent  these 
two  words  by  signs.  By  general  usage  the  sign  (*.•)  is  read 
"since/"  and  the  sign  (.-.)  is  read  ''therefore." 

226.  1.  In  a  town  ^^  of  the  people  are  sick  and  512  are 
well.     How  many  are  sick  ? 

§^  of  the  people  —  -j^-  of  the  people  =  ^|  of  them. 
•.•  ^1  of  the  people  =  512  persons, 
t/s  of  the  people  =  16  persons, 
and  /s  of  the  people  =  48  persons. 

2.  If  electricity  passes  through  7200  miles  of  wire  in  | 
of  a  second,  what  is  its  rate  a  second  ? 

3.  If  a  horse  trots  |  of  a  mile  in  2^  minutes,  in  wliat  time 
can  he  trot  a  mile  ? 

4.  One  half  of  a  post  is  in  the  water,  ^  in  mud,  and  4  ft. 
6  in.  in  the  air.     Find  length  of  the  post. 

5.  After  selling  .32  of  his  slieep  to  one  man  and  .88  of 
them  to  another,  a  drover  had  312  remaining.  How  many 
had  he  at  first  ? 


ILLUSTRATIVE  SOLUTIONS.  151 

6.  A  boy  gave  J  of  his  marbles  to  A,  |  of  the  Remainder  to 
B,  and  thea  had  15  marbles  left.     At  first  he  had  how  many  ? 

After  giving  ^  of  his  marbles  to  A  he  had  ^  of  them  left.     To  B  he 
gave  §  of  this  |,  or  ^,  and  had  ^  —  i,  or  ^  of  them  left. 
.  •.  i  of  his  marbles  =  15,  and  f  of  them  =  90. 

7.  A  man  invested  J  of  his  money  in  land,  J  of  the  re- 
mainder in  cattle,  and  had  $1000  left.  How  much  money 
had  he  at  first  ? 

8.  In  a  certain  school  f  of  the. pupils  belong  to  the  third 
grade,  |  of  the  remaining  pupils  belong  to  the  second  grade, 
and  the  remainder,  which  is  20,  belong  to  the  first  grade. 
How  many  pupils  are  there  in  the  school  ?    ' 

9.  A  man  gives  ^  of  his  property  to  one  son,  J  of  it  to 
another,^  of  it  to  the  third  son,  and  the  remainder,  $550,  to 
his  daughter.     What  is  the  value  of  the  whole  property  ? 

10.  A  person  loses  ^q^  of  his  fortune  and  then  ^^  of  the  re- 
mainder, and  then  |  of  what  he  tlien  had,  and  finds  that  he 
has  $3600  left.     How  much  had  he  at  first  ? 


11.  Find  the  average  weight  of  three  men  weighing, 
respectively,  130  lb.,  145  lb.,  and  175  lb. 

They  together  weigh  130  lb.  +  145  lb.  +  175  lb.  =  450  lb.  .-.  the 
average  weight  of  the  three  men  is  ^  of  450  lb.,  or  150  lb. 

12.  If  130  pupils  attend  school  on  Monday,  126  on  Tues- 
day, 122  on  Wednesday,  125  on  Thursday,  and  122  on 
Friday,  what  is  the  average  attendance  for  the  week  ? 

13.  If  the  temperature  as  indicated  by  a  thermometer  at 
eight  different  times  on  a  certain  day  was,  respectively,  37°, 
36°,  36°,  38°,  40°,  38°,  36°,  and  35°,  what  was  the  average 
temperature  for  the  day  ? 

14.  The  pe?^  capita  indebtedness  of  France  is  $116,  of  Prus- 
sia, $37,  Great  Britain  and  Ireland,  $88 ;  Russia,  $31  ;  Spain, 
$74,  and  the  U.  S.  $15.  What  is  the  average  per  capita 
indebtedness  of  these  six  countries  ? 


152  SCHOOL  ARITHMETIC. 

15.  If  the  heart  beats  140  times  a  minute  during  the  first  3 
years  of  life,  120  times  a  minute  for  the  next  3  years,  100 
times  a  minute  for  the  next  6  years,  90  times  a  minute  for 
the  next  10  years,  and  75  times  a  minute  for  the  next  28 
years,  what  is  the  average  number  of  beats  a  minute  in  a  life 
of  50  years  ? 


16.  The  sum  of  two  numbers  is  84,  and  their  difference  is 
14.     Find  the  numbers. 

The  greater  +  the  less  =  84. 
The  greater  —  the  less  =  14. 
.*.  2  X  the  less  =  70, 
and  the  less  —  35. 

17.  The  sum  of  two  numbers  is  603,  and  their  difference  is 
273.     Find  the  numbers. 

18.  The  difference  between  A's  money  and  B's  is  $16.50, 
and  they  together  have  $166.50.     How  much  has  each  ? 

19.  George  has  J  doz.  eggs  more  than  Kate,  and  both  have 
42  eggs.     How  many  has  each  ? 

20.  The  sum  of  two  numbers  exceeds   their  difference  by 
198.     What  is  the  smaller  number  ? 


21.  A  crew  can  row  60  miles  down  stream  in  3  hours,  but 
requires  4  hours  to  row  back.  What  is  the  rate  of  the  cur- 
rent ? 

Rate  in  still  water  +  rate  of  current  =  20  miles  an  hour. 
Rate  in  still  water  —  rate  of  current  =  15  miles  an  hour. 
.'.  2  X  rate  of  current  =  5  miles  an  hour, 
and  the  current's  rate  =  2|  miles  an  hour. 

22.  A  steamboat  goes  72  miles  down  stream  in  6  hours,  but 
is  8  hours  returning.     What  is  the  rate  of  the  stream  ? 

23.  Going  down  stream  A  can  row  11  miles  in  2  hours. 
Going  up  stream  he  can  row  1^  miles  in  a  half  hour.  Find 
the  rate  of  the  current. 


illustrative:  solutions.  163 

1i4.  If  a  man  can  row  GJ  miles  an  hour  down  stream  and 
4J  miles  an  hour  up  stream,  how  far  can' he  row  in  an  hour  in 
still  water  ?  

25.  A  garrison  of  1200  men  had  provisions  to  last  90  days, 
but  30  days  later  300  more  men  arrived.  How  long  did  the 
provisions  last  after  the  increase  in  the  number  of  men  ? 

They  would  have  lasted  1200  men  60  days, 
or  1  man  1200  x  60  days. 

.-.  they  lasted  1500  men  l??!^-^,  or  48  days. 
1500 

26.  A  garrison  at  Manila,  consisting  of  2500  men,  had  pro- 
visions for  30  days,  but  1500  men  were  withdrawn.  How 
long  did  the  provisions  last  the  remainder  ? 


27.  A  can  do  a  piece  of  work  in  3  days,  and  B  can  do  it  in 
5  days.     How  long  will  it  take  both  together  ? 

In  1  day  A  does  ^  of  the  work. 

In  1  day  B  does  i  of  the  work. 

In  1  day  A  and  B  do  i  +  i,  or  ^g  of  it. 

To  do  if,  or  the  whole  work,  will  require  \^  -j-  ^\-  or  1|  days. 

28.  A  can  do  a  piece  of  work  in  5,  B  in  6,  and  C  in  8  days. 
How  long  would  it  take  them  to  do  it  together  ? 

29.  A  and  B  can  build  a  house  in  3  months.  B  alone  can 
complete  it  in  8  months.  In  what  time  can  A  build  the 
house  ? 

30.  If  A  can  lay  a  certain  wall  in  4|  days,  and  B  in  5^  days, 
how  long  will  it  take  both  together  ? 


31.  Two  pipes  lead  into  a  tank.  One  can  fill  it  in  50  min- 
utes, the  other  in  40  minutes.  There  is  a  discharging  pipe 
which  can  empty  the  tank  in  25  min.  In  what  time  will  the 
tank  be  filled  if  all  three  pipes  are  in  operation  ? 

The  L.  C.  M.  of  50,  40,  and  25  is  200.  Hence  one  pipe  fills  the  tank  4 
times  in  200  min.,  the  other  5  times,  and  both  9  times.  The  third  pipe 
empties  the  tank  8  times  in  200  min. 

.'.  the  tank  is  filled  9  —  8,  or  1  time,  in  200  min.,  or  3  hr.  20  min. 


154  .    SCHOOL  AEITHMETIC. 

82.  One  pipe  will  fill  a  cistern  in  3  hours  ;  a  waste-pipe 
will  empty  it  in  2  Hours.  If  the  cistern  is  full  and  both 
pipes  are  opened,  in  what  time  will  the  cistern  be  emptied  ? 

33.  An  empty  cistern  has  two  pipes.  One  fills  it  in  40 
minutes,  the  other  empties  it  in  60  minutes.  If  both  are 
opened,  in  what  time  will  the  cistern  be  filled  ? 


34*   If  8  horses  eat  48  bushels  of  corn  in  24  days,  in  how 
many  days  will  4  horses  eat  38  bushels  ? 
8  horses  eat  48  bu.  in  24  days. 
1  horse  eats  48  bu.  in  8  x  24  days. 

1  horse  eats  1  bu.  in  — ^^— i-  days. 

48 

.  •.  4  horses  eat  1  bu.  in  — ^ days, 

48  X  4 

and  4  horses  eat  38  bu.  in  ^  ^  24  x  38  ^        ^  ^g  . 

48  X  4  ^  ^ 

35.  If  8  men  in  7  days  can  reap  a  field  of  40  acres,  how 
many  acres  will  be  cut  by  24  men  in  28  days  ? 

36.  If  3  men  earn  $150  in  20  days,  how  many  men  will 
earn  $157.50  in  9  days,  at  the  same  rate  ? 

37.  If  5  yards  of  cloth  f  yd.  wide  cost  $3.12^,  how  much 
will  15  yards  of  that  cloth  1  yd.  wide  cost; at  the  same  rate  ? 

38.  If  a  5-cent  loaf  weighs  1.5  pounds  when  wheat  is  50 
cents  a  bushel,  what  should  it  weigh  when  wheat  is  $.75  a 
bushel? 


39.  When  a  certain  number  is  multiplied  by  9,  the  product 
divided  by  12,  the  qxiatient  increased  by  96,  and  the  sum 
divided  by  3,  the  quotient  is  36.     What  is  the  number  ? 

36  X  3   =108  ;  108-96  =  12; 
12  X  12  =  144  ;  144  -f-    9  =  16. 

40.  If  a  certain  number  is  diminished  by  76,  the  remainder 
multiplied  by  2,  the  product  increased  by  148,  and  the  sum 
divided  by  12,  the  quotient  is  21^.     Find  the  number. 


ILLUSTRATIVE  SOLUTIONS.  155 

41.  At  what  time  between  1  and  2  o^clock  are  the  honr  and 

minute  hands  of  a  clock  together  ? 

First  Solution. 

In  1  hour  the  minute  hand  moves  over  60  minute-spaces,  and  the  hour 
hand  over  5,  Hence  the  former  gains  55  minute-spaces  in  that  time. 
To  gain  1  space  requires  A"  of  an  hour.  At  1  o'clock  the  hands  are  5 
spaces  apart.  Hence  to  gain  tliis  5  spaces  will  require  5  x  ^-  hr.  =  tV 
hr.,  or  5  A"  min.     Hence  the  time  is  5/f  min.  past  1  o'clock. 

Second  Solution. 

The  minute  hand  gains  11  hour-spaces  in  going  12,  that  is,  in  1  hour. 
Hence  to  gain  1  space  requires  -,V  of  an  hour.  At  1  o'clock  the  hands  are 
one  hour-space  apart,  which  the  minute  hand  will  gain  in  I'r  hr.,  or  5^^ 
min.     Hence  it  will  overtake  the  hour  hand  at  5/,-  min.  past  1  o'clock. 

42.  At  what  time  between  4  and  5  o'clock  do  the  hour  and 
minute  hands  of  a  cIocIj:  coincide  ? 

43.  At  what  time  between  10  and  11  o'clock  do  the  hour 
and  minute  hands  of  a  watch  coincide  ? 

44.  At  what  time  between  1  and  2  o'clock  are  the  hands  of 
a  clock  exactly  opposite  each  other  ? 

45.  A  and  13  start  from  the  same  point  and  travel  in  the 
same  direction.  A  goes  7  miles  an  hour  and  B  3  miles  an 
hour.  If  B  has  a  start  of  5  hours,  when  will  he  be  overtaken 
by  A? 


RELATION    OF    NUMBERS. 


227.  The   Relation   of  Numbers  is   their   comparative 
value. 

Thus,  comparing  2  with  4,  we  say  2  is  ^  of  4. 

228.  To  find  a  number  when  part  of  it  is  given. 

1.  82  is  J  of  my  money.     How  much  have  I  ? 

2.  Nettie   spent   $5,  which  was  ^  of   her   money.     How 
much  money  had  she  ? 

3.  Harry  lost  3  marbles,  which  was  ^  of  all  he  had.     How 
many  had  he  ? 

4.  Two  is  J  of  what  number  ?    ^  ?    i  ? 

5.  Five  is  J  of  what  number  ?    i? 

6.  Eight  is  f  of  what  number  ? 


^  of  the  number  =  8. 
.-J"    *^        ^^      =  Jof  8,  or  4. 
.  I  ^^    ''        ''      =  3  X  4,  or  12. 

7.  Nine  is  f  of  what  number  ? 

8.  Ten  is  |  of  what  number  ? 
Find  the  number  of  which 


Since  8  is  f  of  some  number, 
1  third  of  that  number  is  |  of 
8,  or  4  ;  and  since  4  is  1  third 
of  the  number,  3  thirds,  or  the 
number,  equals  3  times  4,  or 
12.     Hence  8  is  f  of  12. 


9.  12  is  f 

17. 

72  is  f  |. 

10.  15  is  f . 

18. 

65  is  If. 

11.  24  is  |. 

19. 

120  is  f . 

12.  29  is  I. 

20. 

217  is  |. 

13.  32  is  j\. 

21. 

210  is  f 

14.  39  is  A. 

22. 

225  is  f  f 

15.  40  is  If 

23. 

414  is  If. 

16.  52isf 

24. 

1000  is  .5. 

RELATION  OF  NUMBERS. 


157 


25.  2  is  f  of  what  number  ? 

f  of  the  number  =  2. 
.i-    -        -      =iof2,  or|. 
p.    ..       "       =  4  X  I,  or  |. 


Since  2  is  J  of  some  number, 
1  fourth  of  that  number  is  i  of 
2,  or  f  ;  and  since  ^  is  i  of  the 
number,  4  fourths,  or  the  num- 
ber, equals  4  times  f,  or  f. 
Hence  2  is  J  of  2J. 


26.  9  is  4  of  what  number  ? 

27.  16  is  f  of  what  number  ? 

28.  A.  lady  spent  $20,  which  was  |  of  her  money, 
much  had  she  at  first  ? 

29.  I  is  f  of  what  number  ? 


How 


^  of  the  number  =  ^. 
'.\  ''    "         ''        =  |of  f,  or  4. 
■.  ^  -    -       ..       ,,  7  X  i  or  3^. 


Since  |  is  f  of  some  number, 
1  seventh  of  that  number  is  i 
of  §,  or  i  ;  and  since  ^  is  |  of 
the  number,  },  or  the  number, 
equals  7  times  ^,  or  V,  or  3Jf. 
Hence  §  is  ?  of  3^ 

30.  ^  is  4  of  what  number  ? 

31.  A  hat  cost  11^,  which  was  f  of  the  cost  of  a  vest.  How 
much  did  the  vest  cost  ? 

32.  William  walked  27  miles  in  one  day,  which  was  J  of 
the  distance  Joseph  walked.     How  far  did  Joseph  walk  ? 

33.  A  lot  cost  $450,  which  was  ^  of  the  cost  of  a  house. 
Find  the  cost  of  the  house. 

34.  Three  fifths  of  the  distance  from  Pittsburg  to  Phila- 
delphia is  213  miles.     How  far  apart  are  the  two  cities  ? 

35.  A  bought  a  horse  for  $126,  which  was  ^  of  what  he 
sold  him  for.     How  much  did  he  gain  ? 

36.  In  fj  of  a  mile  are  216  rods.  How  many  rods  in  a 
mile  ? 

37.  $18  is  f  of  what  B  paid  for  a  cow.  His  horse  cost 
twice  as  much  as  the  cow.  How  much  did  he  pay  for  the 
horse  ? 


158 


SCHOOL  ARITHMETIC. 


ALIQUOT    PARTS. 

229.  The  Aliquot  Parts  of  a  number  are  the  parts  that 
will  exactly  divide  it.     Thus,  2  and  5  are  aliquot  parts  of  10. 
The  relation  of 


50    to  100 

is  J. 

Hence 

50    = 

^  of  100. 

334  to  100 

isi 

i( 

m  = 

i  of  100. 

25    to  100 

isi. 

i( 

25    = 

i  of  100. 

20    to  100 

hi. 

a 

20    = 

1  of  100. 

16|  to  100 

hi. 

<( 

m  = 

I  of  100. 

142  to  100 

hi. 

i  ( 

14f  = 

1  of  100. 

12J  to  100 

isi. 

te 

12^  = 

i  of  100. 

Hi  to  100 

hi. 

te 

1H  = 

^  of  100. 

10    to  100 

isiV- 

i( 

10    = 

3^^  of  100. 

8J  to  100 

isiV- 

{< 

8*  = 

^j  of  100. 

6i  to  100 

iST>,. 

(( 

H  = 

iV  of  100. 

The  numbers  in  the  first  column  are  aliquot  parts  of  100. 
From  them  may  be  found  the  following  other  parts  of  1  ?0  : 
40  =  I  of  100.  37^  =  f  of  100.  83^  =   |  of  100. 

60  =  I  of  100.  62 J  =  1  of  100.  41|  =  ^\  of  100. 

80  =  I  of  100.  87^  =  i  of  100.  5.8^  =  j\  of  100. 

75  =  f  of  100.  66f  =  I  of  100.  31^  =  j%  of  100. 

230.  To  multiply  by  the  aliquot  parts  of  lOO. 

1.  Multiply  2856  by  25. 

4)285600         Since  25  is  i  of  100,  we  multiply  by  100— which  is  done 
WVA()f)     by  annexing  two  ciphers — and  take  ^  of  the  product. 

Multiply  in  a  similar  manner  : 

2.  3576  by  33^. 

3.  2748  by  50. 

4.  1728  by  12f 
6.  3270  by  16f. 

10.  At  $.62 J  a  bushel,  what  must  be  paid  for  1648  bushels 
of  wheat  ? 


6.  4368  by  8^. 

7.  5138  by  14|.  , 

8.  2946  by  66|. 

9.  35768  by  37^. 


RELATION  OP  NUMBERS.  159 

11.  Find  the  cost  of  75  dozen  shovels  at  $.87^  each  ? 

12.  When  potatoes  are  worth  66f^  a  bushel,  how  much 
must  be  paid  for  240  barrels,  each  containing  3^  bushels  ? 

231.  To   divide  by  the  aliquot  parts  of  lOO. 

1.  Divide  1257  by  33^. 
12.57         Since  33^  is  i  of  100,  we  divide  by  100— which  is  done  by 
3     pointing  off  two  decimal    places — and  multiply   the   quotient 

Divide  in  a  similar  manner  : 

2.  2576  by  16f.  7.  3344  by  11^. 

3.  2718  by  25.  8.  76512  by  6^. 

4.  3592  by  12f  9.  37584  by  37.5. 
6.  5340  by  14|.  10.  21360  by  66^. 
6.  4825  by  8^  11.  576900  by  62 J. 

12.  If  pears  are  worth  $.33^  a  peck,  how  many  can  be 
bought  for  $12.50? 

13.  When  butter  is  selling  at  $.16|  a  pound,  how  much 
can  be  purchased  for  $1.50  ? 

14.  At  $^  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  If  ? 

15.  James  can  walk  33^  miles  in  a  day.  How  far  can  he 
walk  in  six  weeks  ? 

16.  A  farmer  has  2400  bushels  of  corn  to  husk.  In  a  day 
he  can  husk  75  bushels.  In  how  many  days  can  he  husk  the 
entire  crop  ? 

17.  In  an  orchard  ^  of  the  trees  bear  apples,  ^  peaches, 
I  pears,  and  the  rest,  5  of  them,  cherries.  How  many 
trees  in  the  orchard  ? 

18.  A  father  willed  f  of  his  estate  to  one  son,  f  of  i-he  re- 
mainder to  another,  and  the  rest  to  his  wife.  If  one  son  re- 
ceived $900  more  than  the  other,  how  much  did  the  widow 
receive  ? 


RATIO. 

232.  To  find  the  relation  of  one  number  to  another. 

1.  2  feet  is  what  part  of  4  feet  ?     Of  6  feet  ? 

2.  How  does  $5  compare  with  $10  ?     With  120  ? 

3.  What  is  the  relation  of  3  to  6  ?     4  to  8  ?     5  to  15  ? 

4.  Howdi3es$10compare  Avith$2  ?  12  with  3  ?  16  with  8  ? 
12  with  4  ? 

6.  What  is  the  relation  of  3  to  8,  or  what  part  of  8  is  3  ? 

Since  1  is  ^  of  8,  3  is  3  times  i  of  8,  or  |  of  8,  Hence  |  is  the  rela- 
tion of  3  to  8. 

6.  What  is  the  relation  of  4  to  9  ?     Of  8  to  12  ?     Of  6  to  14  ? 

233.  The  relation  of  one  number  to  another  of  the  same 
kind  is  Ratio. 

Note. — There  is  no  ratio  between  $3  and  6  hats,  nor  can  the  ratio 
between  ^  feet  and  8  yards  be  determined  in  this  form  ;  but  if  we  reduce 
the  8  yards  to  feet,  the  ratio  is  found  to  be  ^^  or  \. 

234.  The  Sign  of  ratio  is  (  : ). 

Thus,  the  ratio  of  3  to  8  is  written  3  :  8.  This  was  first  used  as  a  sign 
of  division  by  Leibnitz.  The  ratio  of  3  to  8  may  be  expressed  in  three 
ways— 3  :  8,  3  -j-  8,  and  |. 

235.  The  two  numbers  compared  are  together  called  a 
Couplet. 

236.  The  first  is  called  the  Antecedent,  the  second  the 
Consequent. 

Thus,  in  the  ratio  3  :  8,  3  is  the  antecedent,  and  8  the  consequent. 

Notes. — 1.  The  ratio  of  one  number  to  another  is  found  by  dividing 
the  antecedent  by  the  consequent. 

2.  The  ratio  being  the  quotient  of  one  number  divided  by  another  of 
the  same  kind  is  always  an  abstract  number.     (Art.  99.) 


RATIO.  101' 

Since  the  antecedent  may  be  regarded  a3  a  dividend,  and 
the  consequefit  as  a  divisor,  we  have  the  following 

237.  Principles. — 1.  Multiplying  the  antecedent  or  divid- 
ing the  consequent  hy  any  number  multiplies  the  ratio  by  that 
number. 

2.  Dividing  the  antecedent  or  multiplying  the  consequent  by 
any  number  divides  the  ratio  by  that  number. 

3.  Multiplying  or  dividing  antecedent  and  consequent  by 
the  same  number  does  not  change  the  value  of  the  ratio. 

238.  Since  a  ratio  is  the  quotient  of  an  antecedent  by  its 
consequent,  it  follows  that 

(a).  The  antecedent  =  the  consequent  x  the  ratio, 
(b).  The  consequent  =  the  antecedent  ~  the  ratio. 

Find  the  ratio  of  : 


1.  12  to  16. 

9.  5  to  .5.                          "i 

2.  7  to  21. 

10.  .5  to  5. 

3.  8  to  18. 

11.  50^  to  100^. 

4.  $10  to  125. 

12.  $5  to  $2. 

5.  30  to  15. 

13.  52  to  13. 

a  20  yd.  to  30  yd. 

14.  18  to  15. 

7.  27  to  9. 

15.  12  to  17. 

8.  9  to  27. 

16.  95  to  19.                         : 

17.  What  is  the  ratio  of  1  foot  to  a  yard  ? 

18.  What  is  the  relation  of  $.12|-  to  %l  ? 

19.  What  is  the  ratio  of  2.5  to 

.25? 

20.  What  is  the  ratio  of  f  to  f 

? ,            .'-  •;  ■  \i.  .;.*f 

Query. — When  the  denominators  are 

;  the  same,  why  may  they  be  di's-' 

•egarded  ? 

21.  What  is  the  ratio  of  |  to  | 

p 

Suggestion.   ^  -^  |^  =  |  x  f  =  j§,  or 

•i     Or, 

l-^^  =  \%-^\l=\h 

ori 

What  is  the  ratio  of  : 

22.  f  to  f  ?    I  to  i  ?    I  to  I  ? 

23.  I  to  i  ?    i  to  t  ?    4  to  i  ? 

11 


162  SCHOOL  ARITHMETIC. 

What  is  the  fatio  of  ; 

24.  1  to  ^  ?    2  to  I  ?    3  to  I  ? 

25.  i  to  1  ?    i  to  2  ?    f  to  1  ? 

26.  f  to  .7?    6  to  1.2?     10  to2J? 


27.  What  number  has  to  6  the  ratio  3  ?    To  7  the  ratio  5  ? 

28.  Mention  two  numbers  whose  ratio  is  |. 

29.  How  much  is  6  :  2  more  than  7:3? 

30.  Which  is  greater,  f ,  2  -=-  3,  or  1  :  2  ? 

31.  Find  the  value  of  (2  :  6)   x  (5  :  2). 

32.  Find  the  value  of  (5  :  3)  -h  (12  :  f ). 

33.  If  the  antecedent  is  12  and  the  ratio  3,  what  is  the 
consequent  ? 

34.  If  the  consequent  is  5  and  the  ratio  3,  what  is 
the  antecedent  ? 

35.  What  is  the  effect  produced  on  the  ratio  4  :  5  by  multi- 
plying the  antecedent  by  3  ?  Tlie  consequent  by  3  ?  Both 
by  3? 

36.  What  is  the  ratio  of  4«  to  2a  ?     Of  6^>  to  31)  ? 

37.  What  is  the  ratio  of  6d  to  10Z»  ?     Of  7a  to  21«  ? 

38.  AVhat  number  has  to  5  the  ratio  3  ?  What  number  has 
to  a  the  ratio  b  ? 

39.  What  is  the  ratio  oi  b  to  a  ?     Oi  p  to  q  ? 

40.  If  the  antecedent  is  Qa  and  the  ratio  3,  what  is  the 
consequent  ? 

41.  If  a  is  the  antecedent  and  2  the  ratio,  what  is  the 
consequent  ? 


PROPORTION. 

239.  1.  What  is  the  ratio  of  2  to  4  ?  Of  3  to  G  ?  Are 
these  ratios  equal  to  each  other  ? 

2.  Since  tliey  are  equal,  they  may  be  written  thus  : 
2:4  =  3:0. 

This  may  be  read  in  two  ways  :  thus,  the  ratio  of  2  to  4 
equals  the  ratio  of  3  to  G  ;  or,  2  is  to  4  as  3  is  to  G. 

240.  When  two  ratios  are  equal,  and  written  as  above, 
they  form  an  Equality  of  Ratios. 

241.  An  equality  of  ratios  is  a  Proportion. 

Thus,  8  :  4  =  12  :  6  is  a  proportion.  It  may  also  be  written 
8  -T-  4  =  12  -4-  6,  or  I  =  V- 

242.  A  proportion  being  composed  of  two  ratios  must  use 
four  numbers — two  antecedents  and  two  consequents.  When 
any  three  of  these  are  given,  the  fourth  may  be  found. 

Find  the  wanting  number  in  the  following : 

1.  2  :  4  =  5  :  (     ).  6.  3  :  G  =  2  :  (     ). 

2.  1  :  2  =  4  :  (     ).  7. 

3.  2  :  (     )  =  3  :  6.  8. 

4.  1  :  3  =  (     )  :  8.  9. 

5.  (     )  :  5  =  8  :  4.  10.   (     )  :  G  =  3  :  9. 
Queries. — 1.  Which  numbers  are  antecedents  ?    2.  Which  are  con- 
sequents ?    3.  Is  every  proportion  an  equation  ?    Why  ? 

243.  The  sign  (  :  :  )  is  sometimes  written  between  the 
equal  ratios  instead  of  the  sign  of  equality  (  =  ). 

Thus,  3:6=4:8  may'be  written  3  :  6  :  :  4  :  8,  and  is  read  3  is  to  6 
as  4  is  to  8. 


8  : 

4  = 

:    G 

:( 

). 

10 

:( 

)-- 

=  8 

:4. 

4  : 

2  = 

^( 

): 

5. 

164  SCHOOL  ARITHMETIC. 

244.  The  first  and  fourth  numbers  of  the  proportion  are 
the  Extremes,  and  the  second  and  third  the  Means. 

Thus,  in  the  proportion  5  :  10  =  2  :  4,  5  and  4  are  the  extremes,  10 
and  2  the  means. 

245.  The  ratio  of  one  number  to  another  is  a  Simple 
Katio. 

246.  A  Simple  Proportion  is  an  equality  of  two  simple 
ratios. 

247.  Principles. — 1.  77ie  product  of   the  extretnes    is 
equal  to  the  product  of  the  means. 

In  any  proportion,  as  2  :  4  —  3  :  6,  the  ratios  may  be  expressed  in 
fractional  form.     Thus,  2:4  =  3:6  may  be  written  f  =  ^.     Changing 

to  similar  fractions,  we  have  — ^ — -  = ^. 

24  24 

Hence,  2x6  =  3x4.     But  2  and  6  are  the  extremes,  and  3  and  4  the 

means  of  the  given  proportion  ;  therefore,  the  product  of  the  extremes 

is  equal  to  the  product  of  the  means. 

2.  The  product  of  the  means  divided  hy  either  extreme  is 
equal  to  the  other  extreme. 

3.  The  product  of  the  extremes  divided  hy  either  mean  is 
equal  to  the  other  mean. 

^^°  Have  the  pupils  prove  principles  2  and  3. 

Find  the  number  omitted  in  each  of  the  following : 

(  )• 


1.  6  :  8  =  9  :  (  ). 

9.  10  feet  :  15  feet :  :  1 

2.  12  :  10  =  (  )  :  5. 

10.  1  :  6i  :  :  8  :  (  ). 

3.  9  :  15  =  12  :  (  ). 

11.  $14  :  17  :  :  (  )  :  1. 

4.  16  :  (  )  =  18  :  27. 

12.  f  :  2  :  :  5  :  (  ). 

5.  50  :  20  =  (  )  :  12. 

13.  (  )  :  f  :  :  13^  :  If. 

6.  (  )  :  15  =  4|^  :  9. 

14.  3.5  :  ^  :  :  (  )  :  ^. 

7.  6.5  :  19.5  =  i  :  {  ). 

15.  i:  (  )::  f  :  i- 

8.  7.5  :  2.5  =  (  )  :  1.3. 

16.  24  :  (  )  :  :  10  :  J^. 

APPLICATIONS     OF     SIMPLE     PROPORTION. 

248.  Many  problems  that  are  usually  solved  by  analysis 
can  be  readily  solved  by  proportion. 


PROPORTION.  165 

1.  If  8  hats  cost  $10,  how  much  will  12  hats  cost  ? 

First  Statement.  Since  only  like  numbers  can  be  compared 

hats    hats        $  $        in  a  ratio,  we  have  for  the  first  couplet  8 

R    '    12  ■=  K)  '  (      )      ^^^^  ^^^  ^^  hats,  either  of  which  may  be 

-  jj        ^  P  made  the  antecedent.     For  the  second  coup- 

__  24  Ifit  we  have  the  cost  of  8  hats  and  the  cost 

8  of  12  hats.     In  the  first  statement  we  made 

8  hats  the  antecedent,  and  it  is  less  than  the  consequent.  Therefore, 
the  antecedent  in  the  second  couplet  must  be  less  than  the  consequent. 
It  is  evident  that  8  hats  cost  less  than  12  hats  ;  hence  $16  must  be 
made  the  antecedent  in  the  second  couplet.  Solving  by  Principle  2, 
we  find  the  cost  of  12  hats  to  be  $24. 

Second  Statement.  In  the  second  statement  we  make  12  hats 

hats    hats        $           $  the  first  antecedent,  which  is  greater  than 

-j^2   •    8  =  f     )    •  16  ^^^  consequent.     Hence,  the  required  cost, 

^jj        P  which   is  greater  than   $16,  must  be  maile 

• =  24  the  second  antecedent.      Solving   by  Prin- 

*8  ciple  3,  we  get  the  same  answer  as  before. 

It^^  The  pupil  will  note  that  8  hats  bears  the  same  relation  to  12  hats 
as  the  cost  of  8  hats  bears  to  the  cost  of  twelve  hats. 

Note.— In  solving  problems  of  this  kind  in  proportion,  we  apparently 
multiply  hats  by  dollars.  This  is  due,  however,  to  the  form  of  work. 
Since  the  ratio  of  8  hats  to  12  hats  is  equal  to  the  ratio  of  8  to  12,  we  may 
write  8  :  12  =  $16  :  ($    ).     In  practice  this  substitution  is  not  necessary. 

2.  If  5  men  can  build  a  house  in  40  days,  in  how  many 

days  can  8  men  build  it  ? 

It  is  evident  that  8  men  can  do  the 
men    men^     days   ^  days        ^^^.^  .^^  j^^^  ^.^^  ^^^^^  5  ^^^  .  ^^^^^ 

(a)  5     :    8  —    (      )    :    40  ^^^^  required  number  is  less  than  40, 

(b)  8:5=      40     :  (      )        and  the  proportions  are  easily  expressed 

as  in  the  statements. 

Note. — In  example  (a)  we  have  a  direct  proportion.  Thus,  8  hats  : 
12  hats  =  cost  of  8  hats  :  cost  of  12  hats.  In  example  (b)  we  have  an 
inverse  proportion.  Thus,  5  men  :  8  men  =  time  of  8  men  :  time  of  5 
men. 

Query. — How  does  an  inverse  proportion  differ  from  a  direct  propor- 
tion ? 


166  SCHOOL  ARITHMETIC. 

Rule. — For  the  first  couplet  compare  the  two  like  numbers. 
For  the  second  couplet  compare  the  remaining  number  with 
the  required  number.  Determine  from  the  conditions  of  the 
problem  which  is  the  greater. 

Arrange  these  two  numbers  as  a  couplet^  making  the  greater 
or  less  the  antecedent  according  to  the  arrangement  of  the  first 
couplet. 

Divide  the  product  of  the  means  or  extremes  hy  the  sirigle 
mean  or  extreme. 

The  quotient  will  be  the  required  number. 

The  work  may  frequently  be  sliortened  by  cancellation.  This  rule  is 
often  called  "The  Rule  of  Three,"  because  three  numbers  are  given  to 
find  a  fourth. 

3.  If  12  pounds  of  tea  cost  $5,  how  much  must  be  paid 
for  24  pounds  ? 

4.  At  8  cents  a  yard  how  much  will  10  yards  of  calico  cost  ? 
6.  A  tree  60  feet  high  casts  a  shadow  80  feet  long.     How 

long  a  shadow  is  cast  by  a  tree  48  feet  high  ? 

6.  If  18  horses  eat  12  bushels  of  oats  in  a  day,  how  many 
horses  would  eat  20  bushels  in  the  same  time  ? 

7.  If  a  train  runs  9.0  miles  in  3  hours,  how  far  will  it  run 
in  24  hours  ? 

8.  By  working  9  hours  a  day,  B  can  dig  a  ditcli  in  16  days. 
In  how  many  days  can  he  dig  it  by  working  10  hours  a-day  ? 

9.  If  a  family  uses  6  barrels  of  flour  in  10  months,  how 
many  barrels  would  last  a  year  ? 

10.  A  man  earns  $1000  in  6  months.  In  how  many  months 
can  he  earn  $2500  ? 

11.  If  f  of  an  acre  cost  130,  how  much  must  be  paid  for  6 
acres  ? 

12.  A  farmer  raised  3G  bushels  of  wheat  on  \^  acres.  At 
this  rate,  how  many  bushels  can  he  raise  on  10  acres  ? 

13.  In  how  many  days  can  12  men  build  a  wall  that  5  men 
can  build  in  18^  days  ? 


PROPORTION.  167 

14.  Mr.  C  paid  |5l40  for  7  months'  rent.  How  much  did 
he  pay  in  a  year  ? 

15.  How  long  would  it  take  3  men  to  mow  a  field  that  7 
men  can  mow  in  3\  days  ? 

16.  Find  the  cost  of  75  sheep  at  $6.40  each. 

17.  A  paid  115.75  for  3J^  weeks'  board.  At  the  same  rate, 
how  much  did  he  pay  in  a  year  ? 

18.  If  f  of  a  yard  of  cloth  costs  $|,  how  much  must  be  paid 
for  16  yards  ? 

19.  If  10  tons  of  hay  last  5  horses  8  months,  how  long  would 
it  last  12  horses  ? 

20.  If  20  bushels  of  wheat  produce  6^  barrels  of  flour,  how 
many  bushels  will  produce  100  barrels  ? 

21.  If  24  yards  of  carpet  cover  f  of  a  floor,  how  many  yards 
will  cover  the  entire  floor  ? 

22.  If  9  compositors  can  set  up  a  6-page  paper  in  8  hours, 
in  how  many  hours  can  they  set  up  a  20-page  paper  ? 

23.  If  a  cows  cost  b  dollars,  how  much  will  3a  cows  cost  ? 

24.  When  b  hats  cost  c  dollars,  how  much  must  be  paid 
for  d  hats  ? 

25.  Complete  the  equation,  a  :  b  =  7a  :  (     ). 

SUPPLEMENTARY     EXERCISES     (FOR     ADVANCED     CLASSES), 

249.  1.  The  Washington  monument  casts  a  shadow  223 
feet  6^  inches  long  when  a  post  3  feet  high  casts  a  shadow 
14.5  inches.     Find  the  height  of  the  monument. 

2.  Milk  is  worth  20^  a  gallon,  but  by  watering  it  the  value 
is  reduced  to  15^  a  gallon.  Find  the  ratio  of  water  to  milk 
in  the  mixture. 

3.  If  4  is  one  third  of  a  certain  number,  what  is  one  half 
of  it  ? 

4.  Sixty  men  can  grade  a  street  in  40  days.  After  24  days, 
one  third  of  the  men  are  discharged.  In  how  many  days  can 
the  others  finish  the  work  ? 

5.  A  piece  of  work  can  be  done  in  5  weeks  by  12  men.     At 


168  SCHOOL  ARITHMETIC. 


^ 


the  end  of  2  weeks  ifc  is  decided  to  complete  the  work  in  6 
days.     How  many  more  men  must  be  employed  ? 

6.  If  a  4-cent  loaf  weighs  9  ounces  when  flour  is  $6  a  bar- 
rel, how  much  should  a  5-cent  loaf  weigh  when  flour  is  18 
a  barrel  ? 

7.  If  300  cats  kill  300  rats  in  300  minutes,  how  many  cats 
can  kill  100  rats  in  300  minutes  ? 

8.  Two  men  or  three  boys  can  plow  an  acre  in  |  of  a  day. 
How  long  will  it  take  3  men  and  2  boys  to  plow  it  ? 

9.  If  4  horses  or  6  cows  can  be  kept  10  days  on  a  ton  of 
hay,  how  long  will  it  last  2  horses  and  12  cows  ? 

10.  If  8  men  or  15  boys  plow  a  field  in  15  days  of  9^  hours, 
how  many  boys  must  assist  16  men  to  do  the  work  in  5  days 
of  10  hours  each  ? 

11.  Divide  $1000  between  A  and  B  so  that  A  shall  have  $3 
out  of  every  $5. 

There  are  1000  -r-  5  =  200  fives. 
.  •.  A  gets  200  X  $3  =  $600. 
1^"  A  gets  $3  for  every  $2  B  gets.     The  money  is  divided  in  the  ratio 
of  3  to  2. 

12.  Tom  and  Peter  found  a  watch  worth  $45,  and  agreed 
to  divide  the  value  of  it  in  the  ratio  of  f  to  |.  How  much 
was  each  one's  share  ? 

13.  Gunpowder  is  composed  of  nitre,  charcoal,  and  sulphur 
in  the  proportion  of  38,  7,  and  5.  How  many  pounds  of  sul- 
phur in  180  pounds  of  powder  ? 

14.  If  20  men  can  perform  a  piece  of  work  in  12  days,  how 
many  men  can  do  a  piece  of  work  3  times  as  large  in  ^  of  the 
time  ? 

15.  A  man  rides  a  certain  distance  at  the  rate  of  6  miles 
an  hour,  and  walks  back  at  the  rate  of  3^  miles  an  hour. 
If  the  time  of  the  round  trip  is  4f  hours,  what  is  the 
distance  ? 

16.  If  da  sheep  cost  $24,  how  much  will  5b  sheep  cost,  if 
a  :  ^^  :  :  2  :  1  ? 


THE     EQUATION. 

250.  An  Equation  is  a  statemeut  that  two  numbers  or 
expressions  are  equal. 

Thus,  1  +  1  =  2,     7+3  =  2x5,     and  3a;  —  2  =  11  are  equations. 

251.  An  equation  is  like  a  balanced  scale-beam — one  side 
representing  the  thing  weighed, 
and  the  other  side  the  weiglits. 

262.  The  part  of  an  equation 
that  is  written  before  the  sign  = 
is  called  the  first  member,  and 
that  written  after  the  sign  the 
second  member. 

,    Thus,  in  the  equation  7  +  3  =  2x5, 
7  +  3  is  the  first  member,  and  2  x  5  is  the  second  member. 

1.  The  two  members  may  differ  widely  in  form,  but  in  value  they 
must  be  the  same. 

2.  In  the  expression  4  -r-  3  —  2  =  (  ),  the  second  member  is  to  be 
supplied. 

253.  The  numbers  that  compose  the  members  are  called 
the  Terms.  The  terms  of  each  member  are  separated  from 
each  other  by  the  signs  +  or  —  . 

Thus,  in  the  equation  3+6  x  7— 8-f-4  =  43,  the  terms  of  the  first 
member  are  3,  6  x  7,  and  8  -e-  4.  In  the  expression  5+2x4— (  )  = 
10,  one  term  is  to  be  supplied. 

INDUCTIVE    EXERCISES. 

254.  1.  When  will  a  scale-beam  balance  ? 

2.  If  two  one-pound  packages  of  coffee  are  placed  in  one 
scale,  what  must  be  done  to  restore  the  balance  ? 


170  SCHOOL  ARITHMETIC. 

3.  If  one  of  the  packages  is  removed,  what  must  be  done 
to  restore  tlie  balance  ? 

4.  Would  the  balance  be  destroyed  if  the  coffee  and  weight* 
should  change  scales  ?     Why  not  ? 

6.  Could  anything  else  of  equal  weight  be  substituted  for 
the  coffee  without  destroying  the  balance  ? 

6.  If  5  pounds  are  added  to  one  side,  what  must  be  done  to 
the  other  side  to  keep  the  balance  true  ? 

7.  If  one  pound  is  taken  from  one  scale,  how  can  the  bal- 
ance be  restored  ? 

8.  What  does  the  expression  7  4-  5  +  3  =  15  mean  ? 

9.  If  we  add  one  to  the  first  member,  how  much  must  be 
added  to  the  second  to  make  the  sides  equal  ? 

10.  If  we  take  5  from  the  first  member,  what  must  be  done 
to  the  second  to  preserve  the  equality  ? 

11.  Is  there  any  difference  in  value  between  7  +  3  and  15 
-  5  ?     Then  7  +  3  =  15  -  5. 

12.  How  much  is  6  times  (7  +  3)  ?  How  much  is  6  times 
(15  —  5)  ?     Are  the  products  equal  ? 

13.  Then  what  may  be  done  to  both  sides  of  an  equation 
without  destroying  the  equality  ? 

14.  What  is"^  the  quotient  of  (7  +  3)  -f-  2  ?  Of  (15  -  5) 
-^  2  ?     Are  the  quotients  equal  ? 

16.  Then  what  else  may  be  done  to  both  sides  without 
destroying  the  equality  ?  ^ 

16.  A  10-acre  field  is  .worth  $100  an  acre,  and  a  20-acre  field 
is  worth  $50  an  aci'e.    In  what  respects  are  the  two  fields  equal  ? 

255.  Since  one  member  of  an  equation  is  equal  in  value  to 
the  other,  whatever  is  done  to  one  side  must  be  done  to  the 
other  in  order  to  preserve  the  equality. 

256.  Principles. — 1.  Expressions  which  are  equal  to  the 
same  thing  or  to  equal  things  are  equal  to  each  other. 

2.  If  equals  are  added  to  or  subtracted  from  equals,  the  re- 
sults are  equal. 


THE  EQUATION.  171 

3.  If  equals  are  multiplied  or  divided  by  the  same  number ^ 
the  results  are  equal. 

4.  In  general,  if  the  same  operations  are  performed  upon 
both  members  of  an  equation^  the  results  are  equal. 

5.  If  two  expressions  are  equal,  either  can  be  substituted 
for  the  other  ivherever  it  occurs. 

Bt^"  These  principles  are  self-evident,  and  arc  called  axioms. 

CHANGE     OF     FORM. 

267.  It  is  frequently  necessary  to  change  the  form  of  an 
equation  in  order  to  simplify  it.  The  principal  change  is 
transposing  terms. 

268.  Transposiiigr  is  the  process  of  changing  a  term  from 
one  member  of  an  equation  to  the  other  without  destroying 
the  equality. 

1.  In  the  equation  $16  —  $5  =  $8  +  $3,  transpose  15  to 
the  second  member. 

Adding  $5  to  each  member  (Prin.  2)  we  have 

$16  -  $5  +  $5  =  $8  +   $3  +   |5. 
But  since  —  $5  +  f  5  =  0,  we  may  write  the  equation 
$16  =  $8  +13  +  $5. 
How  does  this  equation  compare  with  the  one  given  ? 
It  will  be  noticed  that  (—  $5)  has  disappeared  from  the  first  member, 
while  (+  $5)  appears  in  the  second, 

2.  In  the  equation  2  +  5  =  4  +  3,  transpose  3  to  the  first 
member. 

Subtracting  3  from  each  member  (Prin.  2),  we  have 

2  +  5-3  =  4  +  3-3. 

But  since  +  3  —  3  =  0,  we  may  write  the  equation 

3  +  5-3  =  4. 

How  does  this  equation  compare  with  the  one  given  ? 
It  will  be  noticed  that  3  has  disappeared  from  the  second  member, 
and  appears  in  the  first,  with  a  dififerent  sign  before  it. 

269.  Any  term  may  be  transposed  from  one  member  of 
an  equation  to  the  other  by  dropping  it  from  the  member  iu 


172  SCHOOL  ARITHMETIC. 

which  it  stands,  and  writing  it  in  the  other  with  a  different 
sign. 

Transpose  so  that  only  odd  numbers  will  be  in  the  first 
member  : 

1.  7-4  =  8-5.  4.  5x3-2x4  =  14-6. 

2.  2  +  9  =  1  +  10.  5.  39  -^  3  -  4  ^  2  =  22  -^  2. 

3.  19  +  6  =  30  -  5.         6.  7  +  4  -  8  =  26  -  25  +  2. 

Transpose  so  that  only  like  terms  will  be  in  each  member  : 

7.  3  times  a  number  —  5  =  2  times  the  number. 

8.  5  times  A's  money  —  $10  =  3  times  A's  money  +  $20. 

9.  ^  of  my  money  =  $9  —  |  of  my  money. 

10.  I  of  A's  age  +  4  years  =  :^  of  A's  age  +  11  years. 

11.  $2  +  10  cents  =  II  +  $^  +  60  cents. 

12.  3  rods  +  2  yards  =  2  rods  -I-  7^  yards. 

Transpose  all  terms  containing  x  to  the  first  member,  and 
all  others  to  the  second  member  : 

13.  7a;  -  5  =  3a;  +  15. 

14.  5  +  6a;  -  9  =  2a;  +  8. 
16.  5a;  -i-  6  -  8  =  4  +  3a;. 

16.  9a;  -  36  =  24  -  6a;. 

17.  8a;  -  2y  =  Uy  -  16a;. 

18.  3y  +  10a;  =  10  +  lla;. 

260.  After  like  terms  have  been  collected  into  one  mem- 
ber, the  equation  may  often  be  made  still  more  simple  by 
performing  the  operations  indicated. 

Thus,  the  equation, 

5  times  A's  money  —  3  times  A's  monoy  =  $240  +  $60, 
may  be  written 

2  times  A's  money  : 


Perform  indicated  operations : 

1.  4  times  a  number  —  3  times  the  number  =  10  +  5. 

2.  2  X  B's  money  +  3  x  B's  money  =  $1000  -  $500. 


THE  EQUATION.  173 

3.  7  X  5  -  G4  -^  16  -  11  =  5  X  4.      * 

4.  6  +  5x0  +  18-2  =  2  +  2x25. 
d^"  This  may  be  called  unititig  like  terms. 

Notes, — 1.  In  uniting  terms  we  subtract  the  8um  of  the  minus  terms 
from  the  sum  of  the  plus  terms. 

2.  The  sign  x  may  be  omitted  between  factors  if  one  (or  more)  of  the 
factors  is  a  letter. 

Supply  the  wanting  term  : 

5.  32  -  5  X  3  +  45  -r-  9  =  (     ). 

6.  46  -  18  +  16  +  (     )  =  30  +  7  X  10. 

7.  75  +  13  X  (     )  -  51  =  64  -  2  X  7. 

8.  l  +  ix|-(     )  =  2-2xi 

261.  Expressing  the  conditions  of  a  problem  in  the  form 
of  an  equation  is  called  stating  the  problem.  The  several 
steps  of  the  analysis  may  be  expressed  in  a  series  of  equations, 
each  derived  from  the  one  preceding,  by  a  change  of  form 
under  Principles  1 — 5. 

1.  If  3  melons  are  worth  60  cents,  how  much  are  5  melons 
worth  ? 

1.)  The  cost  of  3  melons  =  60  cents. 
2.)  .*.  the  cost  of  1  melon  =  20  cents. 
3.)  .'.  the  cost  of  5  melons  =  100  cents. 

Queries. — How  is  equation  2  derived  from  1  ?  How  is  3  derived 
from  2  ? 

2.  24  is  f  of  what  number  ? 

1.)     f  of  some  number  =  24. 
2.)  .-.  ^  of  the  number  =  12. 
3.)  .'.  f  of  the  number  =  36. 
Query, — How  are  equations  2  and  3  derived  ? 

3.  Find  f  of  40  ? 

1,)        I  of  forty  =  40.       . 
2,)  .-,  i  of  forty  =  5. 
3.)  .-.  i  of  forty  =  15. 

Query. — How  is  equation  1  derived  ? 


174  SCHOOL  ARITHMETIC. 


^ 


4.  35  is  how  many  eighths  of  40  ? 

40  =  8  eighths  of  40. 
.-.  5  =  1  eighth  of  40. 
.-.  35  =  7  eighths  of  40. 

262.  The  following  solutions  show  a  few  of  the  many  ad- 
vantages of  using  the  equation  in  arithmetic.  Its  utility  in 
solving  many  of  the  problems  of  percentage,  etc.,  appears  in 
later  pages  of  this  book. 

1.  Eight  times  a  number  diminished  by  46  equals  14  more 
than  3  times  the  number.     Find  the  number. 

First  Solution. 
8  times  the  number  —  46  =  3  times  the  number  +  14. 
Transposing  46, 

8  X  the  number  —  3  x  the  number  =  14  +  46. 
Uniting  terms, 

5  X  the  number  =  60. 
.'.  the  number  =  12. 

Second  Solution. 

Let  X  =  the  number. 
.-.  8a;  -  46  =  3a:  +  14. 
Transposing, 

8a:  -  3a;  =  14  +  46. 
Uniting  terms, 
5a:  =  60. 
.'.  a:  =  12. 

1^*"  In  the  solution  of  any  problem,  the  number  to  be  found  is  the  one 
that  must  be  represented  by  a  letter.  This  letter  may  be  treated  just  as 
the  number  itself  would  be  if  known. 

2.  A  coat  and  a  hat  cost  ^3G.  The  coat  cost  5  times  as 
much  as  the  hat.     What  was  the  cost  of  the  hat  ? 

Let  X  —  cost  of  hat. 
5a;  =  cost  of  coat. 
.*.  a:  4-  5a;  =  cost  of  both. 
.*.  6a:  =  $36. 
.-.  x  =  $6,  cost  of  hat, 
and  5a;  =  $30,  cost  of  coat. 


THE  EQUATION.       ,  175 

3.  A  farmer  has  100  hens  and  chicks.  Every  hen  has  9 
chicks.     How  many  of  each  has  he  ? 

Suggestion. — Let  x  =  the  number  of  hens. 

4.  If  three  times  A^s  age  plus  12  years  equals  five  times  his 
age  less  8  years,  what  is  A's  age  ? 

Let  X  =  A's  age. 
3a;  +  12  yr.  =  5x  -  8  yr. 

3a:  -  5a;  =  -  8  yr.  -  12  yr. 
-2a;--20yr., 
or  2a;  =  20  yr. 
a;  =  10  yr. 

U^"  The  signs  of  all  the  terms  can  be  changed  without  destroying 
the  equality  ;  for,  by  transposition,  the  members  can  be  interchanged  and 
therefore  their  signs  changed. 

Queries. — 1.  In  solving  problems,  what  is  the  first  thing  to  do  ? 
(State  the  problem.) 

2.  What  is  the  second  step  ?    The  third  ? 

3.  How  do  you  explain  the  work  after  terms  have  been  united  ?  Let 
the  pupil  write  a  rule. 

1^"  In  solving  the  following  problems  great  care  must  be  exercised 
in  making  the  statement.     Solve  all  by  using  x. 


5.  A  and  B  have  $80,  and  for  each  dollar  B  has,  A  has  $3. 
How  much  has  each  ? 

6.  $40  is  14  more  than  ^  of  my  money.  How  much  have  I? 

7.  18  is  I  of  what  number  ? 

8.  What  number  added  to  ^  of  itself  equals  2x9? 

9.  I  of  a  number  is  5  more  than  f  of  the  number.  What 
is  the  number  ? 

10.  A  lady  bought  a  dress  for  $24,  and  found  that  she  had 
I  of  her  money  left.     How  much  money  had  she  at  first  ? 

11.  Mr.  E  has  gold  dollars  and  silver  dollars  to  the  amount 
of  $30.  He  has  one  half  as  many  silver  dollars  as  gold  dol-. 
lars.     How  many  of  each  has  he  ? 

12.  A  got  \  of  his  father's  fortune,  B  got   I  of  it,  and  C 


176  SCHOOL  ARITHMETIC. 


^ 


got  the  remainder.     If  A  got  $2000  more  than  B,  how  much 
did  0  get  ? 

13.  If  18  is  added  to  6  times  a  certain  number,  the  sum  will 
be  22  less  than  |  of  150.     What  is  the  number  ? 

14.  A  pole  36  feet  long  was  broken  into  two  unequal  pieces, 
f  of  the  longer  piece  being  equal  to  |  of  the  shorter.  How 
long  was  each  piece  ? 

15.  Three  boys.  A,  B,  and  C,  have  77  marbles  ;  B  has  10 
more  than  A,  and  A  has  8  more  than  C.  How  many  has 
each  boy  ? 

16.  Find  a  number  such  that  if  it  be  added  to  ^  of  itself 
the  sum  will  be  6Q.         , 

17.  By  selling  a  watch  for  $36,  I  gained  f  of  its  cost. 
What  was  its  cost  ? 

18.  f  of  A^s  money  diminished  by  $20  is  equal  to  ^  of  his 
money  increased  by  $5.     Find  A's  money. 

19.  A  and  B  together  sold  728  bushels  of  wheat,  and  B 
sold  3  times  as  much  as  A.     How  much  did  each  sell  ? 

20.  Divide  $1000  between  C  and  B,  so  that  C  may  have  | 
as  much  as  B. 


21.  What  number  increased  by  one  half  of  itself,  by  one 
third  of  itself,  and  by  18  more,  will  be  doubled  ? 

22.  If  with  the  money  I  now  have,  I  had  3  times  as  much, 
and  $25  more,  I  should  have  $125.  How  much  money  have 
I? 

23.  Find  a  number  such  that  the  sum  of  its  half  and  its 
third  may  exceed  the  sum  of  its  fourth  and  its  fifth  by  23. 

24.  A  man  left  ^.  of  his  estate  to  his  wife,  -^  for  charity,  | 
to  his  children,  and  $1400  to  his  servants.  What  was  the 
amount  of  his  estate  ? 

25.  A  man  gave  $100  to  his  3  sons ;  to  the  second  he  gave 
twice  as  much  as  to  the  first,  lacking  $8,  and  to  the  third  he 
gave  3  times  as  much  as  to  the  first,  lacking  $15.  How  much 
did  he  give  to  the  first  ? 


THE  EQUATION.  177 

SUPPLEMENTARY     EXERCISES     (FOR     ADVANCED    CLASSES). 

263.  1.  A  drover,  being  asked  how  many  sheep  lie  had^i 
said,  *^  If  to  ^  of  the  number  of  my  flock  you  add  the  num- 
ber 9^,  the  sum  will  be  99|."     How  many  sheep  had  he  ? 

2.  A's  money  added  to  ^  of  B's  money  equals  $2000.  How 
much  money  has  each,  provided  A's  is  to  B's  as  3  to  4  ? 

3.  A  boy  cut  off  one  half  of  the  length  of  his  kite-string. 
'He  then  added  45|  ft.,  and  found  that  the  new  string  was 
^  of  the  original  length.     What  was  this  original  length  ? 

4.  A  lady  being  asked  the  time  of  day  replied  that  J  of 
the  time  past  noon  equaled  f  of  the  time  to  midnight,  minus 
^  of  an  hour.     Wluit  was  the  time  ? 

6.  After  losing  f  of  my  money  I  earned  |5l2,  and  then  spent 
f  of  what  I  had.  What  I  then  had  left  was  $36  less  than  I 
had  at  first.     How  much  had  I  at  first  ? 

6.  A  thief  stole  |  of  A's  money  and  spent  f  of  the  amount 
stolen  before  he  was  caught.  The  remainder,  $324.75,  was 
secured.     How  much  money  had  A  at  first  ? 

7.  A,  B,  and  C  together  have  $434.  f  of  A's  money  equals 
I  of  B's,  and  f  of  B's  equals  f  of  C's.     How  much  has  each  ? 

8.  If  f  of  A's  money  is  to  f  of  B's  as  3  to  4,  and  together 
they  have  $1520,  how  much  has  A  ? 

9.  A,  B,  and  C  together  have  $21950.  fof  A's  money,  |  of 
B's,  and  ^  of  C's  are  all  equal  to  each  other.  How  much  has 
each  ? 

13 


REVIEW    WORK. 


ORAL    EXERCISES. 


264.     1.  A  wagon  cost  150,  which  was  ^  of  the  cost  of 
a  horse.     What  was  the  cost  of  the  horse  ? 

2.  If  three  barrels  of  flour  cost  $14.40,  what  will  2|^  barrels 
cost  ? 

3.  Mr.  A  bought  f  of  a  mill,  and  sold  f  of  it  to  B.     What 
part  of  the  mill  did  he  retain  ? 

4.  Twenty-eight  is  7  tenths  of  how  many  times  4  ? 

6.  A  had  60  acres  of  land ;  he  sold  f  of  it  to  B,  and  |  of 
the  remainder  to  C.     How  many  acres  had  he  left  ? 

6.  Eight  ninths  of  27  are  how  many  times  6  ? 

7.  A  man  gave  7  beggars  $1.60  apiece,  and  had  $8.90  left. 
How  much  had  he  at  first  ? 

8.  If  f  of  a  lemon  cost  f  of  a  cent,  how  much  will  3  lemons 
cost  ? 

9.  A  boy  sold  a  watch  for  $16,  which  was  $4  less  than  f  of 
its  cost.     What  did  it  cost  ? 

10.  A  horse  cost  $180,  which  was  3  times  the  cost  of  a  yoke 
of  oxen.     What  was  the  cost  of  an  ox  ? 

11.  A  has  f  as  much  money  as  B,  and  both  have  $50.     How 
much  has  each  ? 

12.  John  and  Jane  each  has  ^  as  much  money  as  Kate,  and 
all  have  $35.     How  much  has  each  ? 

13.  Two  thirds  of  A's  age  is  |  of  B^s  age,  and  the  sum  of 
their  ages  is  77  years.     What  is  the  age  of  each  ? 

14.  Ada  gave  f  of  her  flowers  to  Eli,  and  had  12  remaining. 
How  many  did  she  give  to  Eli  ? 


REVIEW  WORK.  179 

15.  Mr.  D  gave  $45  for  shovels  at  $1.25  each.  How  many 
dozen  did  he  get  ? 

16.  A  lady  bought  2  bushels  and  3  pecks  of  pears  at  $1.50 
a  bushel.     What  did  they  cost  her  ? 

17.  Mrs.  A  bought  a  dress  for  $G.75.  If  she  paid  $|  a 
yard,  how  many  yards  did  she  buy  ? 

18.  A  farmer  has  a  field  containing  16  acres.  If  he  can 
mow  ^  of  it  in  two  days,  how  many  acres  can  he  mow 
in  a  day  ? 

19.  A  can  dig  a  ditch  in  2  days,  and  B  can  dig  it  in  3  days. 
If  they  work  together,  how  long  will  it  take  them  ? 

How  much  of  it  can  A  dig  in  a  day  ?  How  much  can  B  dig  in  a  day? 
Then  how  much  can  both  dig  ?  How  many  sixths  are  to  be  dug  ?  Then 
how  many  days  will  it  take  them  ? 

20.  A  can  build  a  wall  in  2  days,  B  in  3  days,  and  C  in  4 
days.     How  long  would  it  take  all  three  working  together  ? 

21.  A  and  B  can  cut  a  field  of  wheat  in  6  days,  and  B  alone 
can  cut  it  in  10  days.     In  what  time  can  A  alone  cut  it  ? 

22.  A  farmer  bought  9  sheep  for  $45,  and  sold  them  for 
$14  more  than  |  of  what  they  cost.  Did  he  gain  or  lose,  and 
how  much  ? 

23.  A  boy  bought  apples  at  the  rate  of  6  for  5  cents,  and 
sold  them  at  the  rate  of  5  for  6  cents.  How  much  did  he 
gain  on  each  apple  ?    How  much  on  10  apples  ? 

24.  Tom  is  f  as  old  as  his  mother,  who  was  married  24 
years  ago  at  the  age  of  25.     How  old  is  Tom  ? 

25.  When  hay  is  $13.50  a  ton,  and  coal  is  $8  a  ton,  what 
part  of  a  ton  of  coal  can  be  bought  for  -J  of  a  ton  of  hay  ? 

26.  Jane  bought  a  dozen  oranges,  of  which  she  ate  two, 
and  sold  the  remainder  at  2  cents  apiece,  thereby  gaining  1^ 
cents  on  each  orange  bought.     How  much  did  they  cost  each  ? 

27.  After  spending  ^  of  his  money  for  a  cake,  and  ^  of  it 
for  a  ball  and  bat,  Henry  had  $1.40  left.  How  much  had  he 
at  first  ? 

28.  A  can  build  a  boat  in  4  days,  B  in  5  days,  and  0  in  IQ 


180  SCHOOL  ARITHMETIC. 

days.     If  all  work  together  1  day,  how  long  will  it  take  C 
alone  to  finish  ? 

29.  A  and  B  hired  a  rig  for  $11.  A  used  it  one  day,  and 
B  used  it  two  days.     How  much  should  each  pay  ? 

30.  A  can  do  a  piece  of  work  in  half  a  day,  and  B  can  do 
it  in  f  of  a  day.  How  long  will  it  take  them  if  they  work 
together  ? 

31.  Jack  can  eat  a  loaf  in  1^  days,  and  Jill  can  eat  it  in  2 J 
days.     How  long  will  it  last  hoth  ? 

32.  What  number  is  that  which,  being  increased  by  its  ^, 
its  ■^,  and  its  -^^  Avill  be  doubled  ? 

33.  What  number  will  be  doubled  by  adding  to  it  its  ^,  its 
^,  and  5  more  ? 

34.  Mr.  Kay  is  45  years  of  age,  and  f  of  his  age  is  -^  of 
his  wife's  age.  How  old  was  his  wife  when  she  was  married 
20  years  ago  ? 

35.  A  and  B  earned  $110.  If  A  earned  $10  more  than  B, 
how  much  did  each  earn  ? 

36.  A  man  spends  -J  of  his  money,  and  then  loses  ^  of  the 
remainder  ;  he  then  has  $400.     How  much  had  he  at  first  ? 

37.  In  a  school  f  of  the  pupils  study  arithmetic  ;  ^  of  the 
remainder,  algebra ;  and  the  rest,  or  12,  geometry.  How 
many  pupils  are  there  ? 

38.  A  can  do  a  piece  of  work  in  3  days  ;  B  can  do  the  same 
work  in  4  days  ;  if  A  earns  $2  a  day,  what  does  B  earn  a 
day? 

39.  f  of  my  money  is  4  times  my  week's  wages ;  I  have 
$100.     What  are  my  weekly  wages  ? 

40.  In  traveling  72  miles  a  man  went  f  of  the  distance  the 
first  day,  ^  of  the  distance  the  second  day,  and  the  remainder 
the  third  day.     How  far  did  he  travel  the  third  day  ? 


41.  Owen  is  -f  as  old  as  his  father,  and  f  as  old  as  his 
mother.  If  he  is  18  years  old^  how  old  are  his  father  and 
mother  ? 


kEVIEW  WORK.  181 

42.  I  of  C's  money  is  ^\  of  D's,  and  ^  of  D'a  is  120.  How 
much  money  has  C  ? 

43.  If  Mr.  Fox  were  twice  as  old  as  he  is,  ^  of  liis  age 
would  be  20  years.     What  is  his  age  ? 

44.  It  took  I  of  my  money  to  pay  a  debt ;  I  then  paid  |5l2 
for  a  coat,  which  was  f  of  the  money  I  had  left.  Ilow  much 
had  I  at  first  ? 

45.  f  of  the  sum  of  two  equal  numbers  is  20.  What  are 
the  numbers  ? 

46.  What  number  must  be  added  to  the  difference  between 
J  and  ^  to  make  ^  ? 

47.  The  numerator  of  a  fraction,  whose  value  is  |,  is  20. 
What  is  the  denominator? 

48.  A  has  $3.50,  B  has  $1.25  more  than  A,  and  C  has 
as  much  as  A  and  B.  How  much  money  have  they  to- 
gether ? 

49.  If  a  locomotive  moves  f  of  a  mile  in  ||-  of  an  hour, 
what  is  its  speed  per  hour  ? 

60.  -f^  of  a  farm  is  in  crops,  and  ^V  of  the  remainder  i\ 
woodland.     What  part  of  the  farm  is  woodland  ? 

51.  A  man  plows  f  of  a  field  the  first  day,  and  ^  of  it  the 
second  day.  What  part  of  it  does  he  plow  the  third  day,  if 
he  finishes  on  that  day  ? 

52.  A  miller  keeps  as  toll  -\  of  the  corn  to  be  ground. 
What  is  the  ratio  of  the  toll  to  the  meal  returned  ? 

53.  How  long  will  J  of  a  bushel  of  oats  last  a  horse,  if  2^ 
bushels  last  him  one  week  ? 

54.  By  selling  a  horse  for  $96  I  gain  ^  of  the  cost.  What 
did  the  horse  cost  ? 

55.  James  had  36  cents.  He  lost  |  of  it  and  spent  ^  of  it. 
How  much  has  he  left  ? 

56.  If  f  of  a  cord  of  wood  costs  $3,  how  much  will  ^  of  a 
cord  cost  ? 

57.  A  man  bought  5  bushels  of  wheat  for  $4,  which  was 
•J  of  the  cost.     What  was  the  cost  a  bushel  ? 


182  SCHOOL  ARITHMETIC. 

66.  A  can  do  twice  as  much  work  as  B.  How  many  times 
B's  work  can  both  do  ? 

69.  Daisy  traveled  f  of  lier  journey  by  rail,  f  by  water, 
and  the  remainder,  which  was  18  miles,  by  stage.  How  many 
miles  did  slie  travel  ? 

60.  Preston  worked  IG  days,  and  after  paying  for  a  suit  of 
clothes  with  |  of  his  money  had  $24  left.  How  much  did  he 
receive  a  day  ? 

61.  A  watch  and  chain  cost  $90  ;  the  chain  cost  ^  as  much 
as  the  watch.     What  was  the  cost  of  each  ? 

62.  A,  B,  and  C  can  paper  a  room  in  6  hours,  B  and  C 
can  paper  it  in  10  hours.  In  what  time  can  A  alone  paper 
it? 

63.  If  J  of  a  farm  is  worth  $150  more  than  |  of  it,  what 
is  the  whole  farm  worth  ? 

64.  James  being  asked  how  many  marbles  he  had,  said  he 
had  §  as  many  as  Phil,  and  that  both  together  had  155.  How 
many  had  he  ? 

WRITTEN     EXERCISES. 

265.  1.  A  man  bought  20  acres  of  land  at  $50.25  an  acre. 
He  sold  -^  of  an  acre  to  B,  8f  acres  to  C,  and  the  remainder 
to  D.  If  he  received  $05  an  acre  from  B  and  C,  and  $60  an 
acre  from  D,  how  much  did  he  gain  ? 

2.  How  far  can  a  man  walk  in  3f  hours,  at  the  rate  of  3f 
miles  an  hour  ? 

3.  If  f  of  a  pole  is  in  the  ground  and  24  feet  are  above  the 
ground,  how  long  is  the  pole  ? 

4.  If  there  are  69.16  miles  in  one  degree  of  latitude,  how 
many  miles  are  there  in  34f  degrees  ? 

6.  If  2f  yards  of  cloth  are  required  for  a  pair  of  pants,  f 
of  a  yard  for  a  vest,  and  4  yards  for  a  coat,  how  many  yards 
will  be  left  from  a  piece  of  41  yards,  after  5  suits  have  been 
cut  off  ? 

6.  At  the  rate  of  3  for  10  cents,  what  will  75  dozen  oranges 
cost  ? 


REVIEW  WORK.  183 

7.  A  man  paid  $5  for  sugai-  and  $5  for  coffee.  If  sugar 
was  6:^  cents  a  pound  and  coffee  was  25  cents  a  pound,  how 
many  pounds  of  both  did  he  get  ? 

8.  A  man  can  build  a  fence  in  16  days  by  working  9^ 
hours  a  day.  How  much  longer  would  it  take  him  working 
only  8  hours  a  day  ? 

9.  Find  the  cost  of  6  spoons  when  5  dozen  cost  $32. 

10.  The  quotient  is  16.73,  and  the  divisor  is  8|.  How 
much  must  be  added  to  the  dividend  to  make  150  ? 

11.  In  a  square  rod  there  are  272^  squar6  feet.  How  many 
square  rods  are  there  in  9000  square  feet  ? 

12.  A  jeweler  cut  a  wire  |q  of  an  inch  long  into  11  equal 
parts.     How  many  of  the  parts  were  equal  to  half  an  inch  ? 

13.  Walter  spent  ^  of  his  money  and  ^  more,  then  ^  of 
the  remainder  and  $1  more,  then  J  of  the  remainder  and  $^ 
more,  and  then  had  llf  left.     How  much  had  he  at  first  ? 

14.  A  man  owning  |  of  a  store  sold  ^  of  his  share  for 
$2250.     What  was  the  value  of  the  store  ? 

15.  What  number  is  that  whose  |  exceeds  its  |  by  10  ? 

16.  I  bought  a  horse  and  a  buggy,  paying  ^  as  much  for 
the  buggy  as  for  the  horse.  If  both  cost  $340,  what  did  the 
buggy  cost  ? 

17.  Bought  potatoes  at  $2.62^  and  sold  them  for  $3|^  a 
barrel.  If  my  gain  was  $87.50,  how  many  barrels  did  I 
handle  ? 

18.  Seven  men  dug  a  cellar  in  12f  days.  How  long  would 
it  have  taken  3  men  ? 

19.  If  8  be  added  to  numerator  and  denominator  of  ^, 
will  the  value  of  the  fraction  be  increased  or  diminished,  and 
how  much  ? 

20.  If  5  be  subtracted  from  numerator  and  denominator  of 
4,  how  will  the  value  of  the  fraction  be  affected  ?     Why  ? 

21.  Mr.  S  has  60  hens.  He  sold  f  of  them  to  A,  and  f  of 
the  remainder  to  B.  0  bought  what  remained  at  $.75  a  pair. 
How  much  did  C  pay  ? 


184  SCHOOL  ARITHMETIC. 

22.  Mr.  Willis  sold  3  loads  of  hay  weighing,  with  the 
wagons.  If,  IJ,  and  2J  tons  respectively.  The  empty  wagons 
weighed  |,  f,  and  .9  of  a  ton.  Wliat  was  the  value  of  the 
hay  at  $16.50  a  ton  ? 

23.  A  man  bought  20  sheep  at  $4.75  each,  and  3  horses. 
If  he  paid  $470  for  the  sheep  and  horses,  what  was  the  aver- 
age price  of  the  horses  ? 

24.  What  number  multiplied  by  ^  of  13 j  will  produce  1  ? 

25.  A  can  walk  60  miles  in  12^  hours,  and  B  can  walk  it 
in  15  hours.  If  they  are  60  miles  apart,  and  start  at  the 
same  time  to  walk  toward  each  other,  how  far  apart  will  they 
be  in  an  hour  and  a  half  ? 

26.  Divide  (18J  x  18|)  by  (f -f-  .01),  and  add  (|  of  .001) 
to  the  quotient. 

27.  What  is  the  difference  between  3|^  +  12^  x  2^  and 
(3^  +  12i)  X  2i  ? 

28.  A  can  fence  one  side  of  a  square  in  8  days,  and  B  can 
fence  2  sides  of  it  in  12|-  days.  In  what  time  can  both  to- 
gether fence  the  field  ? 

29.  A  man  left  |  of  his  estate  to  his  first  son,  f  to  his  sec- 
ond son,  and  the  remainder  to  his  daughter,  whose  share 
was  $600  less  than  that  of  the  first  son.  Find  each  one's 
share. 

30.  The  width  of  a  stream  was  measured  at  several  points, 
the  measurements  being  as  follows :  42^  feet,  37^  feet,  35 
feet,  41f  feet,  52^  feet,  and  48.875  feet.  What  is  the  average 
width  ? 

31.  What  number  increased  by  f  of  itself  is  equal  to  1  ? 

32.  Two  fifths  of  my  money  is  gold,  I  of  it  is  silver,  and 
the  remainder  is  paper.  I  have  $8  more  paper  money  than 
silver.     How  much  gold  have  I  ? 

33.  A  watch  and  chain  cost  $125.  If  the  chain  cost  |  as 
much  as  the  watch,  what  was  the  cost  of  each  ? 

34.  Twelve  years  ago,  A  was  ^  as  old  as  B,  but  now  he  is 
I  as  old.     How  old  is  A  ? 


REVIEW  WORK.  185 

35.  A  mother  and  3  cliildren  use  a  pound  of  coffee  in  a 
week.  AVheu  the  motlier  is  absent,  two  pounds  last  the  chil- 
dren 6  weeks.  How  long  would  a  pound  last  the  mother 
alone  ? 

36.  If  3  be  added  to  |  of  a  certain  number,  and  J  of  the 
sum  be  multiplied  by  3  tenths  of  3,  the  product  will  be  3 
times  10.8.     What  is  the  number  ? 

37.  A  man  gives  .lof  his  income  to  charity,  .24  for  edu- 
cating his  children,  .375  for  other  expenses,  and  lays  by  the 
remainder,  which  is  $570.     What  is  his  income  ? 

38.  A  man  sold  a  horse  for  1^  times  the  cost,  gaining  $15. 
Find  the  cost. 

39.  I  of  f  of  what  number  equals  -^  of  3G0  ? 

40.  How  much  will  49  men  earn  in  17^  days  @  $2.10  t^ 
day? 

41.  If  T^  of  a  ream  of  paper  costs  $.60,  how  much  can  be 
purchased  for  $237.60  ? 

42.  For  what  length  of  time  will  $495  pay  rent  at  the  rate 
of  $24.75  a  month  ? 

43.  The  greater  jf  two  fractions  is  J|  ;  their  difference  is 
■^\.     What  is  the  other  fraction  ? 

44.  How  many  oranges  @  $.25  per  dozen  will  pay  for  36 
bu.  of  coal  worth  8-^  cents  a  bushel  ? 


45.  A  lady  sold  -^,  f ,  and  f  of  her  fowls  ;  she  had  30  re- 
maining.    How  many  had  she  at  first  ? 

46.  A  farmer  sold  65  bu.  more  than  ^  of  his  wheat,  and 
found  that  the  remainder  was  65  bu.  more  than  f  of  hi  ; 
wheat.     How  many  bushels  luid  he  at  first  ? 

47.  A  drover  spent  $50.60,  wliich   was  ^4  of  what  lie  ii<' 
for  23  sheep.     What  did  he  get  apiece  for  them  ? 

48.  Bought  eggs  at  the  rate  of  If^  each,  a*  d  sold  them  at, 
27^  a  dozen.  I  gained  $3.65  ;  how  many  dozens  did  1 
sell  ? 


186  SCHOOL  ARITHMETIC. 

49.  A  merchant  had  163|  yd.  of  gingham,  from  which  he 
sold  95^  yd.  The  remainder  was  made  into  25  aprons  of  the 
same  size.     How  many  yards  did  each  apron  contain  ? 

50.  A  woman  bought  a  shawl  for  $9.75,  which  was  ^  of  f 
of  the  price  asked.  At  how  much  less  than  tlie  price  asked 
did  she  buy  the  shawl  ? 

61.  John  earns  I24.66|  per  week  ;  James  earns  f  as  much, 
and  Tom  |  as  much  as  John  and  James  together.  If  Tom 
gives  ^  of  his  money  to  charity,  how  much  has  he  left  each 
week  ? 

62.  -j^  of  the  troops  engaged  in  a  battle  were  killed  ;  f  of 
f  of  the  killed  numbered  45.  What  was  the  original  number 
of  the  troops  ? 

63.  If  one  horse  consumes  3|  bu.  of  oats  per  week,  how 
many  bushels  will  18  horses  consume  in  6  weeks  ? 

64.  A  can  do  a  piece  of  work  in  7^  days,  A  and  B  can 
both  do  the  same  work  in  5  days.  In  what  time  cau  B  do 
the  work  ? 

66.  A  lady  bought  a  watch  for  $81.75,  paying  ^  down. 
How  much  must  she  pay  per  month  to  pay  the  remainder  in 
8  months  ? 

66.  Mr.  Drift  sold  f  of  his  property,  ^  of  the  remainder, 
and  -^  of  that  remainder.  His  property  was  then  worth 
$2760.     What  was  the  value  of  the  whole  property  ? 

67.  A  and  B  raised  320  bushels  of  wheat.  After  paying  ^ 
of  it  for  rent,  they  divided  the  remainder  so  that  A  received 
I  as  much  as  B.     How  many  bushels  did  each  receive  ? 

68.  William  bought  330  lemons  at  30  cents  a  dozen,  and 
6^  dozen  oranges  at  2  cents  apiece  ;  he  sold  f  of  the  lemons 
at  4  for  25  cents,  the  remainder  at  3  cents  apiece  ;  he  sold  ^ 
of  the  oranges  at  cost,  the  remainder  at  30  cents  a  dozen. 
How  much  did  he  gain  ? 

69.  A  jeweler  sold  a  watch  for  $62.50,  which  was  ^  more 
than  he  paid  for  it.     How  much  did  it  cost  ? 

60.  Mr.  D  purchasing  a  pair  of  horses  paid  ^  of  the  price 


REVIEW  WORK.  187 

in  cash ;  i  the  remainder  he  paid  a  month  later.     He  still 
owes  $310  of  the  debt.     What  was  the  purchase  price  ? 

61.  Two  men  do  a  piece  of  work  in  18  days.  If  the  first 
man  works  three  times  as  fast  as  the  second,  in  liow  many 
days  can  each  do  the  work  alone  ? 

62.  A  father  left  a  fortune  to  his  three  children  ;  the  old- 
est was  to  have  f  of  it,  the  second  was  to  have  ^  of  it,  and 
the  youngest  was  to  have  the  rest.  The  oldest  received 
$12000  less  than  the  youngest.  What  was  the  value  of  the 
estate  ? 

63.  If  f  of  a  bushel  of  corn  be  worth  f  of  a  bushel  of 
wheat,  and  wheat  be  worth  $1.40  a  bushel,  how  many  bushels 
of  corn  can  be  bouglit  for  $27  ? 

64.  Bought  304  pounds  of  rice  for  $31. IG.  One  fourth  of 
it  is  destroyed.  I  sell  the  remainder  at  a  loss  of  1  cent  on 
the  pound.     What  is  my  whole  loss  by  the  transaction  ? 

65.  Y^Q  of  the  cost  of  my  house  and  barn  is  ?j|  of  the  dif- 
ference between  their  costs ;  and  1^  times  the  difference 
between  their  costs  is  $3600.     Required  the  cost  of  each. 

66.  A  agreed  to  hoe  |  of  a  field  of  corn  while  B  hoed  the 
remainder.  After  finishing,  it  was  found  that  A  had  hoed 
69^  rows  more  than  one-half.     How  many  rows  in  the  field  ? 

67.  The  total  Indian  population  on  the  reservations  in 
1893  was  249,366,  while  the  area  of  the  reservations  was 
134,176  square  miles.  What  was  the  average  quantity  of 
land  occupied  by  100  Indians  ? 

68.  The  moon^s  diameter  is  ^V  ^^^^^  ^f  ^^^^  earth,  and  the 
sun's  diameter  is  110  times  that  of  the  earth.  What  fraction 
of  the  sun's  diameter  is  that  of  the  moon  ? 

69.  I  bought  21  pigs  and  15  cows  for  $693.  Each  cow 
cost  $27  more  than  each  pig  cost.     Find  cost  of  each  pig. 

70.  Three  Indians  counted  their  trophies.  Red  Cloud  had 
twice  as  many  as  Bigfoot,  and  the  latter  had  f  as  many  as 
Horsehead,  who  had  4  less  than  Red  Cloud.  How  many  had 
each  ? 


188  SCHOOL   ARITHMEfid. 


SUPPLEMENTARY     EXERCISES       (FOR     ADVANCED     CLASSES). 

206.  1.  One  day  a  man  spent  f  of  his  money,  and  the  next 
day  4  of  the  remainder.  If  he  had  ^q  of  a  dollar  left,  how 
much  did  he  spend  the  second  day  ? 

2.  A  can  roof  a  house  in  6^  days,  but  can  work  only  .75  of 
each  day.  If  B  helps  him,  the  time  required  to  roof  the 
house  is  2  days  and  f  of  an  hour.  Counting  9  hours  a  day, 
in  how  many  days  can  B  alone  do  the  work  ? 

3.  A  farmer  has  5  horses  to  each  of  which  he  gives  ^  bushel 
of  oats  three  times  a  day.  When  oats  are  worth  $|  a  bushel, 
what  will  be  the  cost  of  the  oats  required  in  the  month  of 
September  ? 

4.  Two  men,  starting  from  the  same  point,  walk  along  a 
railroad  track.  When  one  has  gone  -^  of  a  mile  and  the 
other  f  of  a  mile,  how  many  feet  are  they  apart,  there  being 
1G|  feet  in  a  rod,  and  320  rods  in  a  mile  ? 

5.  If  telegraph  poles  are  ^^  of  a  mile  apart,  how  many 
poles  in  a  line  40.5  miles  long  ? 

6.  When  0  is  a  factor,  what  is  the  product  ?     Why  ? 

7.  When  the  multiplicand  is  one  greater  than  the  product, 
which  is  ^g,  what  is  the  multiplier  ? 

8.  A  man  divided  $10  equally  among  5  boys.  Each  boy 
gave  -^  of  his  share  to  the  poor.  How  much  was  given  to 
charity  ? 

9.  A  man  divided  $a  equally  among  b  boys.  Each  boy 
gave  ^  of  his  share  to  the  poor.  How  much  was  given  to  the 
poor  ? 

10.  At  one  store  a  lady  spent  |-  her  money  and  $^  more  ;  at 
anotlier,  ^  the  remainder  and  $j  more  ;  at  another,  |  the  re- 
mainder and  $1  more,  and  then  had  a  dollar  left.  How  much 
had  she  at  first  ? 

11.  In  an  orchard  there  are  a  rows  of  trees,  and  between 
each  two  rows  of  trees  are  5  rows  of  cabbage.  What  is  the 
value  of  the  cabbage  at  $b  a  row  ?     ($5ab  —  5b.) 


COMPOUND    NUMBERS. 

267.  A  concrete  number  expressed  in  two  or  more  de- 
nominations (units  of  measure)  is  called  a  Compound  Num- 
ber. 

Thus,  2  feet  3  inches,  3  pounds  8  ounces,  are  compound  numbers. 
But  3.5  pounds  and  $1.55  are  not  compound  numbers. 

268.  A  Measure  is  a  unit  by  wbich  quantity,  such  as 
value,  length,  weight,  etc.,  is  estimated. 

Thus,  the  yard  is  a  measure,  because  it  is  a  unit  by  which  length  is 
estimated  or  measured. 

269.  A  Prime  (or  principal)  Unit  is  a  unit  of  measure 
from  which  other  units  of  the  same  kind  may  be  derived. 

Thus,  a  dollar  is  a  prime  (or  principal)  unit. 

MEASURES  OF  VALUE. 

270.  The  ordinary  measure  of  value  is  Money,  which  is 
sometimes  called  currency.  Coin  is  metal  money  ;  all  other 
currency  is  called  Paper  Money. 

271.  United  States  Money  is  the  legal  currency  of  the 
United  States.     The  prime  (or  principal)  unit  is  the  dollar. 

Table. 

10  mills    =  1  cent  (ct.  or  ^). 
10  cents  =  1  dime  (d.). 
10  dimes  =  1  dollar  ($). 

$       d.        ct.         m. 
1  =  10  =  100  =  1000 


190  SCHOOL  ARITHMETIC. 

1.  The  U.  S.  now  issues  the  following  coins: 

Gold.— The  20-dollar,  lO-doUar,  5-dollar,  and  2^dollar  pieces. 
Silver. — The  dollar,  half-dollar,  quarter-dollar,  and  the  dime. 
Nickel. — The  5-cent  piece. 
Bronze. — The  one-cent  piece. 

2.  The  standard  weight  of  the  gold  dollar  is  25.8  gr. ;  it  contains  23.22 
grains  of  pure  gold,  but  is  not  now  coined. 

3.  The  silver  dollar  weighs  412|  gr. 

4.  The  standard  purity  of  the  gold  and  silver  coins  is  9  parts  pure 
metal  and  1  part  alloy  (by  weight).  The  alloy  of  silver  coin  is  pure 
copper.  The  alloy  of  gold  coins  is  copper,  or  silver  and  copper.  If  both 
are  used,  the  silver  is  not  to  exceed  i\,  of  the  alloy. 

5.  The  mill  has  never  been  coined  ;  it  is  merely  a  convenient  name  for 
the  tenth  part  of  a  cent. 

272.  E]nglish  Money  is  the  currency  of  Great  Britain. 
The  principal  unit  is  the  pound  sterling,  which  is  called  the 
sovereign  when  coined. 

Table. 

4  farthings  —  1  penny  (d.).    " 
12  pence       =  1  shilling  (s.). 
20  shillings  =  1  pound  (£). 

=  $4.8665  in  U.  S.  money. 

£       s.        d.         far. 
1  =  20  =  240  =  960. 

273.  Canada  has  a  decimal  currency,  and  the  table  and 
denomihations  are  the  same  as  those  of  U.  S.  money. 

274.  France  also  has  a  decimal  currency.  The  unit  is  the 
franc. 

The  value  of  a  franc  in  U.  S.  money  is  $.193.  The  franc  is  divided 
into  100  centimes. 

275.  German  Money  is  the  legal  currency  of  the  German 
Empire.     The  unit  is  the  mark. 

The  value  of  a  mark  in  U.  S.  money  is  $.2385.  The  mark  is  divided 
into  \QQ  pfennigs. 


COMPOUND  NUMBERS.  191 

1.  How  many  dimes  in  $3  ?     In  $i  ?     In  $3^  ? 

2.  How  many  cents  in  $10  ?  How  many  dimes  ?  How 
many  half-dollars? 

3.  How  many  pence  in  3  shillings  ?  In  8  shillings  ?  In  ^ 
shillings  2  pence  ? 

4.  How  many  pence  in  a  pound  ?    In  £^  ?    In  £2  5s.  ? 

5.  How  many  shillings  in  £5  ?     In  £2  10s.  ?     In  £i  ? 

6.  Have  you  changed  the  value  of  these  numbers,  or  their 
form  f 

276.  Reduction  is  the  process  of  changing  the  denomina- 
tion of  a  number  without  changing  its  value ;  that  is,  chang- 
ing the  unit  of  measure  and  the  number  of  the  units. 

277.  Reducing  a  number  to  a  lower  denomination  is  called 
Reduction  Descending. 

WRITTEN     EXERCISES. 

1.  Reduce  £5  8s.  7d.  to  pence. 

p,        rt^      ^  j-^  Since  there  are  20s.  in  1  pound,  in  £5 

-K\f\       ,  *o~        -i^c  there  are  5  times  20s.,  or  100s.;  100s.  + 

100s.  +  8s.  =  108s.  _        ,.„  ,,  101  •    1   u-iT 

108        -i.^i    _ioq/>i         8s.  =r  108s.    •.*  there  are  12a.  nil  shilhng, 

1  -laaA     X   ^^'a  ~—  ^'ic\'iA       in  108s.  there  are  108  times  12d.,  or  1296d. ; 
129bd.  +  7d.  _  IdOdd.     ^296d.  +  7d.  =  1303d. 

Reduce  to  farthings : 

2.  6s.  4d.  2  far.  5.  9s.  6d.  1  far. 

3.  10s.  lOd.  3  far.  6.  £2  8s. 

4.  13s.  lid.  7.  £3  4s.  9d.  3  far. 

8.  Reduce  £^  to  pence. 

9.  Reduce  £f  to  shillings  and  pence. 

£|  =  I  of  20s.  =  -Lf  a  s.  =  12is. 
is.  =  i  of  12d.  =  6d. 
.•.£§  =  12s.  6d. 
This  may  be  called  reducing  to  loiver  denominations. 

Reduce  to  lower  denominations  : 

10.  £f.  12.  fs.  14.  £f.  16.  2is. 

11.  £3%.  13.  fs.  15.  £lf.  17.  IJs. 


192  SCHOOL   ARITHMETIC. 

18.  Reduce  .36  of  a  shilling  to  lower  denominations. 

.36  of  a  shilling  =  .36  of  12d.  =  4.32d. 
.3-3  of  a  penny  =  .32  of  4  far.=  1.28  far! 
J  .-.  .36s.  =  4d.  1.28  far. 

Reduce  to  lower  denominations  : 

19.  .37s.  21.  £A5  23.  £.1^.  25.  .56s. 

20.  .70S.  22.  £.(JQ  24.  £.|.  26.  .87s. 

278.  1.  In  70  cents  how  many  dimes  ? 

2.  How  many  dollars  in  200  cents  ?     In  500  ct.? 

3.  How  many  shillings  in  24  pence  ?     In  36d.? 

4.  In  GOs.  how  many  pounds  ?     In  £100  ?     In  £70  ? 

5.  Have  you  changed  the  value  of  these  numbers  ?  What 
have  you  changed  ? 

279.  Reducing  a  number  to  a  higher  denomination  is 
called  Reduction  Ascending. 

WRITTEN     EXERCISES. 

1.  Reduce  6845  farthings  to  pounds,  shillings,  etc. 
4  far.)6845  far.  Since  in  1  penny  there  are  4  far.,  in 

^2(]        )1711d   +  1  far.      ^^^'^  ^'^^'  ^^ere  are  as  many  pence  as  4  far. 

rt^  X    -,  ..^ — ~7~rn is  contained  times  in  0845  far.,  or  1711, 

20s.       )   142s.  +  7d.  .^,  1^1^ 

with  a  remainder  ot  1  lar. 

^^   +  -^s.  Since  there   are    12d.  in  1  shilling,   in 

1711d.  there  are  as  many  shillings  as  12d.  is  contained  times  in  1711d., 
or  142,  with  a  remainder  of  7d. 

Since  there  are  20s.  in  1  pound,  in  142s.  there  are  as  many  pounds  as 
20s.  is  contained  times  in  142s.,  or  7,  with  2s.  remaining. 

Therefore,  6845  far.  =  £7  2s.  7d.  1  far. 

Another   Method. 
1  far.  =  id. 

.-.  6845  far.  =  6845  x  id.=  1711id.=  1711d.  +  1  far. 
Id.  =  -,Vs. 
.-.  1711d.=rl711  X  -,\s.=  142i^2S.=  142s.+  7d. 

.-.  142s.  =  142  X  £  iff  =  £"-A-  =  £7  +  2s. 
, ',  6840  far.  =  £7  2s.  7d,  1  far. 


COMPOUND  NUMBERS.  1^ 

Eeduce  to  higher  denominations  : 

2.  7500  far.  4.  8927d.  6.  13785s. 

3.  6738  far.  6.  5360d.  7.  23456  far. 

8.  What  part  of  a  pound  is  ^d.  ? 

(a)  (b) 

Id.  =  -,^js.  i-*-12=5V. 

.-.  id.=  i  of  THrS.=  ^iS.  1^4  +  30  =T^. 

Note. — This  is  really  reducing  ^d.  to  a  higher  denomination,  but,  as 
fractions  are  involved,  it  is  customary  to  say  it  is  reducing  id.  to  the 
fraction  of  a  £. 

Eeduce  to  the  fraction  of  a  £  : 
9.  id.  11.  f  far.         13.  .35d.  15.  -^  far. 

10.  ifar.  12.  |d.  14.  .65  far.         16.  ^^^d. 

17.  AVhat  decimal  part  of  a  pound  is  12s.  8d.  3  far.? 

'  The  explanation  of  this  solu- 

8.75  -^  12  =  .72916  +  ^.j^j^  jg  similar  to  that  given  in 

12.72916  -V-  20  =  £.636458  +  example  8. 

or  The  second  method  is  simple 

£-^  __  9(3Q  f^j.^  and    convenient.      Reduce  both 

12s.  8d.  3  far.  =  611  far.  numbers  to  farthings,  and  find 

611far.  =  £fi^  =  £.  636458+  *^^  ^^^^«- 

18.  Reduce  9s.  6d.  1  far.  to  the  decimal  part  of  a  £. 

19.  Reduce  £5  to  U.  S.  money. 

20.  Reduce  18s.  9d.  3  far.  to  shillings. 

21.  How  many  francs  in  $19.30  ? 

22.  What  is  the  value  in  English  money  of  $243,325  ? 

23.  Reduce  1000  francs  to  U.  S.  money. 

24.  How  many  marks  in  $2385  ? 

25.  Reduce  ^  of  a  pound  to  lower  denominations. 

26.  Reduce  Is.  4d.  to  the  fraction  of  a  pound. 

27.  Express  8  dimes,  7  cents,  5  mills  as  the  decimalpart 
of  a  dollar.  . 

13 


194  SCHOOL  ARITHMETIC. 

MEASURES  OF   CAPACITY. 

280.  Ijiqiiid  Measure  is  used  in  measuring  liquids  of 
all  kinds.     The  principal  unit  is  the  gallon. 

Table. 

4  gills  (gi.)  =  1  pint  (pt.). 
2  pints  =  1  quart  (qt.). 

4  quarts       =  1  gallon  (gal.), 
gal.     qt.     pt.     gi. 
1  r=  4  =  8  =  32. 

1.  In  estimating  capacity,  31|  gal.  are  counted  a  barrel,  and  63  gal. 
a  hogshead  ;  but  in  commerce  they  are  not  fixed  measures. 

2.  The  gallon  contains  231  cubic  inches. 

281.  Dry  Measure  is  used  in  measuring  grain,  fruit, 
vegetables,  etc.     The  principal  unit  is  the  hushel. 

Table. 

2  pints  (pt.)  =  l^quart  (qt.). 

8  quarts  =  1  peck  (pk.). 

4  pecks  =  1  bushel  (bu.). 

bu.    pk.     qt.      pt. 

1  :^  4  =  32  =  64. 

1.  The  standard  bushel  of  the  United  States  contains  2150.42  cubic 
inches. 

2.  Grain,  seeds,  small  fruits,  etc.,  are  sold  by  even  or  stricken  meas- 
ure. Coal,  corn  in  the  ear,  coarse  vegetables,  etc.,  are  sold  by  heaped 
measure. 

3.  In  dry  measure  a  quart  contains  (2150.42  -^  32),  or  67.2  cubic 
inches.     In  liquid  measure  a  quart  contains  (231  -?-  4),  or  57.75  cu.  in. 

ORAL    EXERCISES. 

282.  1.  How  many  quarts  in  16  pt.?  In  28  pt.?  In 
19  pt.? 

8.  How  many  gallons  in  20  qt.  ?     In  32  qt.  ?     In  18  qt.  ? 
3.  What  is  a  gallon  of  milk  worth  at  2  cents  a  gill  ? 


COMPOUND  NUMBERS.  195 

4.  When  vinegar  is  40  cents  a  gallon,  how  much  must  be 
paid  for  2  qt.  ?     1  pt.  ? 

5.  At  5  cents  a  pint,  how  many  gallons  of  molasses  can  be 
bought  for  $2  ? 

6.  How  many  pecks  in  24  qt.  ?     In  35  qt.  ? 

7.  In  2  pecks  how  many  pints  ?     In  3^  pk.? 

8.  How  many  pecks  in  5  bushels  ?     In  12^  bu.  ? 

9.  A  man  bought  12  pk.   of  potatoes  at  10  cents  a  half- 
peck.     How  much  did  he  pay  for  them  ? 

10.  Harry  sold  half  a  bushel  of  chestnuts  at  5  cents  a  pint. 
How  much  did  he  get  for  them  ? 

WRITTEN    EXERCISES. 

283.  Reduce  to  lower  denominations  : 

1.  5  gal.   3  qt.   1  pt.  6.  10  gal.   2  qt.   1  pt. 

2.  f  gal.  6.  .875  gal. 

3.  5  bu.  2  pk.   3  qt.  7.  3  bu.   6  qt. 

4.  %  bu.  8.  .675  pk. 

284.  Reduce  to  higher  denominations  : 

1.  1235  qt.  (liquid  measure).     4.  2457  qt.  (dry  measure). 

2.  1323  qt.  (dry  measure).  6.  501  pt.  (dry  measure). 

3.  321  pk.  6.  1620  pt.  (dry  measure). 


7.  What  part  of  a  gallon  is  f  of  a  pint  ? 

8.  Reduce  2  qt.  1  pt.  to  the  fraction  of  a  gallon. 

9.  What  decimal  part  of  a  gallon  is  1  pt.  ? 

10.  Mrs.  E  bought  2^  gal.  of  milk  at  4  cents  a  pint.     How 
much  did  she  pay  for  it  ? 

11.  What  part  of  a  bushel  is  a  pint  and  a  half  ? 

12.  Hiram  feeds  his  horse  12  qt.  of  oats  in  a  day.     How 
long  will  5  bu.  last  ? 

13.  If  a  bushel  of  salt  is  worth  1.40,  how  much  must  be 
paid  for  4  qt.  ? 

14.  When  apples  are  worth  $1.60  a  bushel,  how  many  pecks 
can  be  purchased  for  $2.40  ? 


196 


SCHOOL  ARITHMETIC. 


MEASURES    OF   WEIGHT. 

285.  Weight  is  the  measure  of  quantity  estimated  by  the 
scale  or  balance  with  reference  to  some  fixed  unit. 

286.  Avoirdupois  Weigiit  is  used  in  weighing  all  coarse 
and  heavy  articles,  such  as  cattle,  horses,  coal,  grain,  grocer- 
ies, and  all  metals  except  gold  and  silver.  It  is  the  common 
comynercial  weight. 

Table. 

16  ounces  (oz.)         =:  1  pound  (lb.). 

100  pounds  =  1  hundred- weight  (cwt.). 

20  hundred- weight  =  1  ton  (T.). 

T.    cwt.      lb.  oz. 

1  =  20  r=  2000  =  32000. 

1.  The  avoirdupois  pound  contains  7000  grains. 

2.  The  long  ton  of  2240  lb.  has  almost  gone  out  of  use  in  the  United 
States. 

3.  A  barrel  of  flour  weighs  196  lb. 


LEGAL 

WEIGHT 

OF 

A 

BUSHEL 

t 

5 
p 

:d 

c 

l-H 

03 

o 

X 

^ 

6 

1 

"S 
§ 

6 

"6 

1-5 

12; 

O 

48 
50 
56 
32 
60 
56 
60 
45 
60 

O 

46 
42 
56 
36 
60 
56 
60 

60 

c 

48 
48 

32 

m 

56 
62 

«3 
> 

48 
52 
56 
32 
60 
56 
60 
45 
60 

i 
^ 

Barley 

50 

32 
54 

60 

48 
48 
5l 
32 
60 
56 

60 

48 

^ 

32 
60 
56 
60 
45 
60 

48 
50 
56 
32 
60 
56 
60 
45 

m 

48 
52 
56 
32 
60 
56 
60 
45 
60 

56 
56 
32 
60 
56 
60 
45 
60 

48 
48 
56 
32 
60 

45 
60 

48 
48 
56 
32 
60 
56 

45 
60 

48 
48 
56 
32 
60 
56 
60 
45 
60 

48 
52 
56 
32 
60 
56 
60 
45 
60 

26 
56 

48 
50 
56 

60 

64 
45 
60 

48 
48 
58 
32 

56 
60 
44 
60 

'18 

Buckwheat 

5>, 

Corn 

56 

Oats 

3'>, 

Potatoes 

60 

Rye 

56 

Clover-seed 

60 

Timothy-seed 

45 

Wheat 

60 

COMPOUND  NUMBERS.  197 

ORAL    EXERCISES. 

287.  1.  In  4000  lb.  how  many  tons  ?  In  7500  lb.?  In 
35  cwt.  ? 

2.  Find  the  cost  of  2  T.  15  cwt.  of  hay  at  116  a  ton. 

3.  What  will  2-^  lb.  of  cinnamon  cost  at  2  cents  an  ounce  ? 

4.  At  $.32  a  pound,  what  will  7.5  oz.  of  butter  cost  ? 
6.  What  part  of  a  pound  is  ^  oz.  ? 

6.  What  decimal  part  of  a  pound  is  12  ounces  ? 

WRITTEN     EXERCISES. 

288.  Reduce  : 

1.  3  T.  4  cwt.  to  oz.  4.  5|  T.  to  lb. 

2.  5  cwt.  45  lb.  to  oz.  5.  7f  cwt.  to  lb. 

3.  6  cwt.  90  lb.  to  lb.  6.  625  lb.  to  oz. 

Reduce  to  higher  denominations  : 

7.  475  oz.  9.  2075  lb.  11.  150  cwt. 

8.  1220  oz.  10.  1000  oz.  12.  75386  oz. 

13.  Reduce  f  of  an  ounce  to  the  fraction  of  a  pound. 

14.  Reduce  12  cwt.  to  tlie  decimal  part  of  a  ton. 

15.  How  many  grains  in  3  lb.  ?     In  5  1b.?    4oz.  ? 

16.  How^  many  barrels  of  flour  in  686  lb.? 

17.  When  hay  is  worth  $15  a  ton,  how  much  must  be  paid 
for  250  lb.  ? 

18.  How  much  will  3  lb.  6  oz.  of  butter  cost,  at  $.32  a 
pound  ? 

19.  If  cloves  are  worth  40  cents  a  pound,  how  much  must 
be  paid  for  40  ounces  ? 

20.  A  man  bought  1500  lb.  of  wheat,  at  $.80  a  bushel. 
What  did  it  cost  him  ? 

21.  Mr.  B  bought  a  bag  of  oats  weighing  112  lb.  If 
oats  were  worth  40  ct.  a  bushel,  how  much  did  he.  pay  for 
them  ? 

289.  Troy  Weig:lit  is  used  in  weighing  gold,  silver,  and 
jewels,  and  in  laboratory  tests. 


198  SCHOOL  ARITHMETIC. 

Table. 

24  grains  (gr.)      =  1  pennyweight  (pwt.). 
20  pennyweights  —  1  on  nee  (oz.). 
12  ounces  —  1  pound  (lb.). 

lb.      oz.      pwt.        gr. 
1  =  12  =  240  =  5760. 

1.  The  carat  is  a  weight  used  in  weighing  diamonds  and  other  jewels. 
It  is  equal  to  3.168  troy  grains. 

2.  The  term  carat  is  also  used  to  indicate  the  fineness  of  gold,  and 
means  2-4  part  of  the  mass.  Thus,  gold  that  is  16  carats  fine  is  ^^  gold, 
and  2%  alloy. 

WRITTEN     EXERCISES. 

290.  1.  Reduce  .625  of  an  ounce  to  the  fraction  of  a 
pound. 

2.  Reduce  f  gr.  to  the  fraction  of  an  ounce. 

3.  What  decimal  part  of  a  pound  is  5  oz,  4  pwt.  ? 

4.  How  many  grains  in  3  lb.  troy  ?     In  3  lb.  avoirdupois  ? 

5.  How  many  rings  can  be  made  from  6  oz.  of  gold,  if  each 
ring  contains  80  gr.  ? 

6.  When  silver  is  worth  1.60  an  ounce,  how  much  must  be 
paid  for  1000  gr.  ? 

7.  When  a  ring  is  marked  "  18  carats/^  what  part  of  it  is 
pure  gold  ? 

8.  How  many  knives,  each  weighing  3  oz.,  can  be  made 
from  5  lb.  9  oz.  of  silver  ? 

9.  At  $.80  a  pwt.,  how  much  must  be  paid  for  a  chain 
weighing  1  oz.  8  pwt.  12  gr.  ? 

10.  What  is  the  present  market  value  of  an  ounce  of 
gold  ?  What  is  the  market  value  of  an  ounce  of  silver  ? 
What  is  the  ratio  of  the  market  value  of  silver  to  that  of 
gold  ? 

291.  Apothecaries'  Weig'lit  is  used,  in  mixing  medicines 
and  in  selling  them  at  retail.  Drugs  in  bulk  are  bought  and 
sold  by  avoirdupois  weight. 


COMPOUND  NUMBERS.  199 


Table. 


20  grains  (gr.)  =  1  scruple  {^). 

3  scruples  —  1  dram  ()  3  . 

8  drams  =  1  ounce  (  3  ). 

12  ounces  =  1  pound  (lb). 

ft).      3         3         3  gi-. 

1  =  12  =  96  =  288  =  5760. 

COMPARISON     OF    WEIGHTS. 
Avoirdupois.  Tboy.  Apotukcaries'. 

1  lb.  =  7000  gr.   1  lb.  =  5760  gr.   1  lb.  =  5760  gr. 

1  oz.  =  437i  gr.   1  oz.  =  480  gr.    1  oz.  =  480  gr. 

It  will  be  observed  that  the  troy  pound  and  ounce  are  identical  with 
the  apothecaries'  pound  and  ounce.  It  will  also  be  observed  that,  while 
the  avoirdupois  pound  is  heavier  than  the  troy  pound,  the  troy  ounce 
is  heavier  than  the  avoirdupois  ounce. 

Queries.— 1.  What  is  the  ratio  of  the  troy  pound  to  the  avoirdupois 
pound  ?  2.  What  is  the  ratio  of  the  troy  ounce  to  the  avoirdupois 
ounce  ? 

Note. — In  compounding  liquid  medicines,  druggists  make  use  of  the 
following  :  GO  minims  (tti)  =  1  dram,  8  drams  =  1  ounce,  16  ounces  = 
1  pint. 

MEASURES  OF  EXTENSION. 

292.  Extension  has  three  dimensions,  called  length, 
breadth,  and  thickness,  or  height. 

A  line  has  but  one  dimension  —  length. 
A  surface  has  two  dimensions  — length  and  breadth. 
A  solid  has  three  dimensions  —  length,  breadth,  and  thick- 
ness. 

293.  Measures  of  Extension  are  used  in  measuring 
lengths,  surfaces,  and  solids. 

294.  Linear  Measure  is  used  in  measuring  length.  The 
principal  unit  is  the  yard. 


200 


SCHOOL  ARITHMETIC. 


Table. 

12  inches  (in.)  =  1  foot  (ft.). 

3  feet  =  lyard  (yd.). 

5^  yards  ) 

or      V         =1  rod  (rd.). 
le^feet    ) 

320  rods  =  1  mile  (mi.), 

mi.      rd.         yd.  ft.  in. 

1  =  320  =  1760  =  5280  =  63360. 


Supplementary  Table. 


3  feet 

4  inches 
18  inches 

21.888  inches 
6  feet 
3  sizes 

8  furlongs 

9  inches 
1  cable  length 

1.15  statute  miles 


=  1  pace. 

=  1  hand  (used  to  measure  height  of  horses) 

=  a  cubit. 

=  1  sacred  cubit. 

=  1  fathom. 

=  1  inch  (used  by  shoemakers). 

=  1  mile. 

=  1  span. 

=  120  fathoms. 

=  1  nautical,  or  geographic,  mile. 
1  nautical  mile  =  1  knot  =  6086  feet. 
3  nautical  miles  =  1  league. 

WRITTEN     EXERCISES. 


295.  Reduce  the  following  : 


3  yd.  2  ft.  to  in. 
1  mi.  40  rd.  to  rd. 
^  mi.  to  feet. 
10  yd.  1  ft.  9  in.  to  in. 
^  rd.  to  inches. 


6.  2  rd.  4  yd.  6  in.  to  in. 

7.  3520  yd.  to  mi. 

8.  7920  ft.  to  mi. 

9.  3000  yd.  to  mi.,  etc. 
10.  720  rd.  to  miles. 


Reduce  to  lower  denominations  : 

11.  -|rd.  13.  .8  yd.  15.  3i  yd. 

12.  I  mi.  14.  .375  mi.  16.  7  K).  2 


COMPOUND  NUMBERS.  201 

17.  Reduce  f  of  a  yard  to  the  fraction  of  a  rod. 

18.  What  part  of  a  rod  is  9  inches  ? 

19.  What  part  of  a  yard  is  2  ft.  8  in.  ? 

20.  At  $.55  a  rod,  how  much  must  be  paid  for  digging  a 
ditch  10  rd.  7  ft.  6  in.  long  ? 

21.  When  cloth  is  worth  90  cents  a  yard,  how  much  must 
be  paid  for  a  piece  16  inches  long  ? 

22.  How  many  pounds  (troy)  in  11520  grains  ? 

23.  How  many  powders  of  gr.  V  each  can  be  made  from  5  6 
of  calomel  ? 

24.  A  ring  which  weighs  120  grains  is  16  carats  fine.  What 
is  the  value  of  the  gold  in  it  at  $20  an  ounce  ? 

296.  Boundaries  of  solids  are  called  Surfaces.  A  surface 
has  length  and  breadth,  but  no  thickness. 

Note. — We  usually  think  of  a  solid  as  a  material  body — e.g.,  a  block 
of  wood — but  the  portion  of  space  occupied  by  the  block  of  wood  is 
regarded  as  the  solid. 

297.  Two  lines  proceeding  from  the  same  point  form  an 
Angle.     The  size  of  the  angle 
depends   upon   the    degree    of 
opening  between  the  lines.  ^^^^  anglk. 

298.  A  flat  surface  bounded  by  straight  lines  or  by  a 
curved  line  is  called  a  Plane  Figure. 

The  distance  around  a  plane  figure  is  called  the  perimeter. 

299.  A  plane  figure  that  has  four  straight  sides  is  called 
a  Quadrilateral. 

300.  When  the  line  CD 
meets  AB,  as  in  the  figure, 
making  the  angles  ACD  and 
DCB  equal,  each  of  these 
angles  is  called  a  Right  .  /- 
Angle.                                        ^  / 


D 


:^) B 


202 


SCHOOL  ARITHMETIC. 


A   RECTANGLE. 


301.  A  quadrilateral  all  of  whose 
angles  are  right  angles  is  called  a 
Rectangle. 

302.  An   equilateral  (equal-sided)    rectangle   is   called   a 
Square. 

1.  Each  angle  is  a  right  angle. 

2.  How  does    the  length  of  a  square  compare 
with  its  breadth  ? 


A 
SQUARE. 


SQUARE   INCH. 


303.  When  each  side  of  a  square  is 
one  inoh  long,  the  figure  is  called  a 
square  inch. 

1.  What  is  Hi  square  foot?  K  square 
yard  ? 

2.  W\\2it  \^  2i  square  tinit  9 

3.  Is  a  square  also  a  rectangle  ? 

304.  The  Area  of  a  figure  or  surface  is  the  number  of 
square  units  of  measure  it  contains.  The  area  is  often  called 
the  superficial  contents, 

A  rectangle  4  feet  long  and  3 
feet  wide  may  be  divided  into  3 
strips,  each  1  foot  wide,  and  each 
strip  into  4  equal  parts,  each  part 
being  1  square  foot  (square  unit  of 
measure). 

Then  the  area  of  the  strip  AB  is 
4x1  sq.  ft.,  and  the  total  area  is 
3  X  4  X  1  sq.  ft.,  or  12  sq.  ft. 

305.  Prixciple. — The  area  of  a  rectangle  is  expressed  hy 
the  product  of  the  numbers  that  represent  its  length  and 
Ireadth. 

Both  dimensions  must  be  expressed  in  like  units. 

1.  A  board  is  8  feet  long  and  1  foot  wide.  How  many- 
square  feet  in  its  surface  ?  How  many  would  there  be  if  it 
were  2  feet  wide  ? 


B 


COMPOUND  NUMBERS.  203 

2.  A  box  lid  is  6  inches  long  and  4  inches  wide.  How 
many  square  inches  in  its  surface  ?  What  unit  of  measure 
is  used  here  ? 

3.  How  many  square  feet  in  the  surface  of  a  square  table 
whose  sides  are  2  feet  ?    3  feet  ? 

4.  How  many  square  inches  in  a  square  whose  sides  are  13 
inches  ?    What  may  this  square  be  called  ?     Why  ? 

5.  How  many  square  feet  in  a  square  whose  sides  are  3 
feet  ?     What  may  this  square  be  called  ?     Why  ? 

6.  The  sides  of  a  square  are  16^  feet.  How  many  square 
feet  in  its  surface  ?     What  may  this  square  be  called  ?     Why  ? 

SQUARE  MEASURE. 

306.  Square  Measure  is  used  in  measuring  surfaces. 

Table. 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 

9  square  feet  =  1  square  yard  (sq.  yd.). 

30|^  square  yards  =  1  square  rod  (sq.  rd.). 

160  square  rods  =  1  acre  (A.). 

640  acres  =  1  square  mile  (sq.  mi.). 

sq.  mi.  A.        sq.  rd.         sq.  yd.  sq.  ft.  sq.  in. 

1  =  640  =  102400  =  3097600  =  278:8400  =  4014489600. 

1.  In  square  measure  the  term  perch  is  sometimes  used  instead  of 
square  rod. 

2.  In  roofing,  slating,  etc.,  100  sq.  ft.  is  called  a  square. 

3.  When  we  say  a  surface  is  a  "  foot  square."  we  mean  that  each 
dimension  is  a  foot.  Hence  the  terms  "foot  square,"  "rod  square," 
etc.,  mean  dimensions,  while  "square  foot,"  "  square  rod,"  etc.,  mean 
area. 

ORAL    EXERCISES. 

307.  1.  A  board  is  2  feet  long  and  J  ft.  wide.  What  is 
its  area  ?    Is  it  square  ? 


204  SCHOOL  ARITHMETIC. 

2.  What  is  the  area  of  a  board  4  yards  long  and  ^  yd. 
wide  ? 

3.  A  lot  is  20  rods  long  and  8  rods  wide.  How  many  sq. 
rd.  in  the  lot  ?  What  may  this  rectangle  be  called  ?  What 
other  dimensions  might  it  have,  and  yet  have  the  same  area? 

4.  What  is  a  square  mile  ?  How  many  rods  long  is  one 
side  of  it  ?  Then  how  can  we  find  how  many  sq.  rd.  in  a 
square  mile  ? 

5.  How  can  we  find  the  number  of  acres  in  a  square  mile  ? 
Name  the  unit  of  measure. 

6.  How  do  you  know  that  there  are  9  sq.  ft.  in  a  square 
yard  ?     Explain  by  a  diagram. 

WRITTEN    EXERCISES. 

308.  Reduce  to  lower  denominations  : 

1.  2  sq.  ft.  12  sq.  in.  5.  2  A.  80  sq.  rd. 

2.  3  sq.  yd.  8  sq.  ft.  6.  3  yd.  2  ft.  8  in. 

3.  5  sq.  yd.  7  sq.  in.  7.  1  A.  25  sq.  yd. 

4.  10  sq.  rd.  5  sq.  yd.  8.  1  sq.  mi.  280  A. 

Reduce  to  higher  denominations  : 

9.  1000  sq.  ft.  11.  3000  sq.  yd.  13.  10000  sq.  rd. 

10.  2000  sq.  in.         12.  4000  sq.  rd.  14.  33333  sq.  in. 

15.  Reduce  |  sq.  rd.  to  square  yards,  etc. 

16.  What  part  of  an  acre  is  |  of  a  square  rod  ? 

17.  One  square  yard  is  what  part  of  a  square  rod  ? 

18.  AVhat  part  of  a  square  rod  is  5  sq.  yd.  8  sq.  ft.  ? 

19.  A  floor  is  16  ft.  long  and  12  ft.  wide.  How  many 
square  feet  does  it  contain  ? 

20.  A  town  lot  is  40  ft.  wide  and  120  ft.  long.  What  is  its 
area  in  square  yards  ? 

21.  How  many  sq.  yd.  in  a  piece  of  carpet  18  ft.  long  and 
a  yard  wide  ? 

22.  A  field  40  rods  long  and  25  rods  wide  contains  how 
many  acres  ? 


COMPOUND  NUMBERS.  205 

23.  At  $50  an  acre,  liow  much  must  be  paid  for'12  A.  32 
sq.  rd.  of  land  ? 

24.  When  a  product  and  one  factor  are  given,  how  may 
the  otlier  factor  be  found  ? 

25.  A  two-acre  lot  is  20  rods  long.     How  wide  is  it  ? 

26.  A  field  containing  10  acres  is  32  rods  wide.  What  is 
its  length  ? 

27.  AVhat  will  it  cost  to  paint  a  ceiling  16  ft.  by  12  ft.,  at 
$.25  a  square  yard  ? 

28.  A  gable  barn-roof,  65  ft.  long  and  30  ft.  wide,  was 
covered  with  sheet-iron.     How  many  sq.  yd.  were  required  ? 

29.  The  area  of  a  rectangular  glass  is  30  sq.  ft.,  and  its 
length  is  144  inches.     What  is  its  width  ? 

30.  A  tract  of  prairie  land  is  6  miles  long  and  4^  miles 
wide.  How  many  farms  of  160  acres  each  could  be  sold 
from  it  ? 

31.  A  lot  60  feet  square  has  in  its  central  part  a  reservoir 
18  feet  square.  What  is  the  distanice  from  the  reservoir  to 
the  fence  surrounding  the  lot  ? 

32.  If  steel  rails  weigh  24  lb.  a  foot,  and  can  be  bought 
for  130  a  ton,  what  will  be  the  cost  of  the  rails  for  a  mile  of 
single-track  railway  ? 

33.  A  man  who  owns  a  farm  120  rods  square  put  10  acres 
in  corn,  15  in  rye,  20  in  oats,  15  in  barley,  and  the  re- 
mainder in  wheat.  What  part  of  the  whole  farm  did  he  put 
in  wheat  ? 

PARALLELOGRAMS. 

How   does  the    figure 

ABCD     differ     from     a       A. . ^B 

rectangle  ?    Are  its  sides 
equal  ?    Are  its  opposite 
sides   equal  ?     Are   they 
parallel  ?     Are  its  angles  D 
equal  ? 


206 


SCHOOL  ARITHMETIC. 


309.  A  quadrilateral  whose  opposite  sides  are  parallel  is 
called  a  Parallelogram. 

Query.— Is  a  rectangle  a  parallelogram  ?    Why  ? 

310.  The  side  of  a  figure  on  which  it  is  supposed  to  stand 
is  called  the  Base ;  as  CD.  The  Altitude  is  the  perpen- 
dicular distance  between  the  base  (or  the  base  extended)  and 
the  side  or  angle  opposite. 

1.  Any  side  may  be  re- 
garded as  the  base. 

2.  What  is  the  diago- 
nal of  a  parallelogram  ? 

(«).  If  we  cut  off  the    ^       ^  D 

end  CAE,  and  place  it  at  the  other  end,  on  DBF,  will  the 
size  of  the  figure  be  changed  ? 

(h).  Then  is  the    area  A  B 

of  ABFE  equal  to  that  of 
ABDC  ?  Is  EF  equal  to 
CD  ? 

(c).  How   is    the   area  c 
of  the   rectangle   ABFE 
found  ?    Then  how  may  the  area  of  the  parallelogram  ABDC 
be  found  ? 

Since  a  parallelogram  is  equivalent  to  a  rectangle  having 
the  same  base  and  altitude,  it  follows  that 

311.  Pri  NCiPLE. —  The  area  of  a  parallelogram  is  expressed 
hy  the  product  of  the  numbers  that  represent  its  base  and 
altitude. 

The  base  and  altitude  of  all  figures  must  be  expressed  in  like  units. 
Find  the  area  of  the  following  parallelograms  : 

1.  Base  24  ft.,  alt.  8  ft.  4.  Base  16  ft.,  alt.  9  in. 

2.  Base  35  ft.,  alt.  15  ft.  5.  Base  42  rd.,  alt.  5^yd. 

3.  Base  72  ft.,  alt.  39  ft.  6.   Base  132  in.,  alt.  30  ft. 

7.  A  board  in  the  form  of  a  parallelogram  is  6  ft.  6  in. 


E 


COMPOUND  NUMBERS. 


207 


long  on  each  side,  and  1   ft.  4  in.  wide.     Draw  the  figure, 
and  find  .the  area. 

8.  The  area  of  a  parallelogram  is  518  sq.  in.  If  its  length 
is  37  inches,  what  is  its  altitude,  or  width  ? 

9.  A  ten-acre  field  is  in  tlie  form  of  a  parallelogram.  Tlie 
shortest  distance  from  one  side  to  the  opposite  side  is  25 
rods.     What  is  the  length  of  the  field  ? 

10.  Mr.  A  has  a  rectangular  field  40  rd.  long  and  20  rd. 
wide.  Mr.  B  has  a  field  of  equal  size  in  the  form  of  a  paral- 
lelogram. If  its  length  is  32  rd.,  what  is  the  shortest  dis- 
tance across  the  field  ? 

TRIANGLES. 

How  many  sides 
has  the  figure  ABC  ? 
Are  they  all 
straight  ? 

312.  A  plane  figure  that  has  three  straight  sides  is  called 
a  Triangle. 

The  point  where  two  sides 
meet  is  called  a  Vertex  ;  as  A. 

(a).  Into  how  many  equal 
triangles  does  the  diagonal  AB 
divide  the  parallelogram 
BCAD  ? 

(b).  Then  the  area  of  each  triangle  is  what  part  of  the  area 
of  the  parallelogram  ? 

(c).  Since  the  area  of  the  parallelogram  is  expressed  by 
AD  X  BE,  how  is  one  half  its  area,  or  the  area  of  the  triangle 
ABD,  expressed  ? 

313.  Principle.  —  The  area  of  a  triangle  is  expressed  by 
one  half  the  product  of  the  numbers  that  represent  its  base 
and  altitude. 

Note. — When  one  angle  of  a  triangle  is  a  right  angle,  the  triangle  is  said 
to  be  right-angled.     In  such  a  triangle  one  side  is  the  altitude,  or  height. 


208  SCHOOL  ARITHMETIC. 

Find  the  area  of  the  following  triangles  : 

1.  Base  12  ft.,  alt.  7J  ft.      3.  Base  47  rd.,  alt.  165  ft. 

2.  Base  29  ft.,  alt.  16  ft.      4.  Base  18  in.,  alt.  12  ft.  6  in. 

6.  The  base  of  a  triangular  piece  of  slate  is  27  inches,  and 
the  altitude  33  inches.  Draw  the  figure,  and  find  the  area  in 
square  feet. 

6.  How  many  triangles,  base  5  ft.  and  alt.  4  ft,,  are  equal 
to  a  parallelogram  whose  base  is  50  ft.,  and  whose  altitude  is 
40  ft.  ? 

7.  A  triangle  whose  base  is  7  yards  has  an  area  of  819  sq. 
ft.     What  is  its  altitude  ? 

8.  A  house  is  24  ft.  wide,  and  the  ridge  of  the  roof  is  12 
ft.  above  the  upper  floor.  Find  the  cost  of  painting  the 
gables,  at  1.40  a  square  yard. 

9.  What  is  the  area  of  a  table  a  feet  square  ? 

10.  A  rectangle  is  a  ft.  long  and  J)  ft.  wide.  What  is  its 
area  ? 

11.  The  altitude  of  a  parallelogram  is  a  ft.,  and  the  base 
is  I  ft.     What  is  the  area  ? 

12.  The  base  of  a  triangle  is  2^  inches,  and  the  altitude  is 
a  inches.     Find  its  area. 

13.  What  is  tlie  area  of  a  square  whose  perimeter  is  4^ 
inches  ? 

14.  One  side  of  a  rectangle  is  %b  ft.,  and  the  perimeter  is 
6J  ft.     What  is  the  area  ? 

TRAPEZOIDS. 

In  the  figure  ABCD  which  sides       ^ ^ •• ■ — f 

are   parallel  ?      How  many   sides      /              !  y  / 

has  the  figure  ?  / i    \ / 

314.  A  quadrilateral  having  two  sides  parallel  is  called  a 
Trapezoid;  as  ABCD. 

(«).  Cut  two  equal  trapezoids  out  of  paper,  and  place  them 
end  to  end  so  as  to  form  a  parallelogram. 


COMPOUND  NUMBERS.  209 

(b).  How  does  the  area  of  one  trapezoid  compare  with  that 
of  the  parallelogram  ? 

315.  If  the  trapezoid  ABCD  in  the  figure  swings  about 
the  point  P,  to  the  position  ECBF,  then  the  whole  figure 
AFED  is  a  parallelogram.  Now  the  area  of  ABCD  is  one- 
half  of  that  of  AFED  ;  but  the  area  of  AFED  is  equal  to 
DE  X  BH.  But  DE  =  DC  +  CE  =  DC  +  AB.  .  •.  the  area 
of  ABCD  =  J  of  (DE  X  BlI)  =  i  of  (AB  +  CD)  x  BH. 
Hence  the  area  of  a  trapezoid  equals  half  the  area  of  a 
parallelogram  having  the  same  altitude  and  a  base  equal  to 
the  sum  of  the  two  parallel  sides. 

316.  Principle. — The  area  of  a  trapezoid  is  expressed  by 
one  half  the  product  of  the  numbers  that  represent  its  altitude 
and  the  sum  of  its  parallel  sides. 

Find  the  area  of  the  following  trapezoids  : 

1.  Altitude  8  in.,  parallel  sides  12  in.  and  10  in. 

2.  Altitude  13  ft.,  parallel  sides  19  ft.  and  11  ft. 

3.  Altitude  25  rd.,  parallel  sides  45  rd.  and  35  rd. 

4.  Altitude  76  ft.,  parallel  sides  6  rd.  and  23  yd. 

6.  How  many  sq.  ft.  in  a  board  12  ft.  long,  15  in.  wide  at 
one  end  and  9  in.  at  the  other  ? 

6.  The  parallel  sides  of  a  trapezoidal  lot  are  75  ft.  and  55 
ft.  respectively,  and  the  shortest  distance  between  them  is  40 
ft.     Draw  the  figure  and  find  the  area  of  the  lot. 

7.  One  side  of  a  farm  in  the  form  of  a  trapezoid  is  135  rd. 
long,  the  side  parallel  to  it  is  121  rd.  long,  and  the  per- 
pendicular distance  between  them  is  100  rd.  What  is  the 
value  of  the  farm  at  $75  an  acre  ? 

8.  A  twenty-acre  field  is  in  the  form  of  a  trapezoid  whose 
parallel  sides  are  72  rd.  and  88  rd.  respectively.  What  is  the 
altitude  ? 

9.  The  distance  around  a  triangular  farm  whose  sides  are 
equal  is  480  rd.,  and  the  altitude  is  80  rd.  How  many  acres 
in  the  farm  ? 

14 


210. 


SCHOOL  ARITHMETIC. 


E 


^c^riSES^^ 


CIRCLES. 

Is  the  portion  of  the  page  en- 
closed by  ADBE  a  plane  figure  ? 

What  kind  of  line  bounds  the 
figure  ? 

Is  any  part  of  the  line  nearer 
the  center  than  another  part  ? 

317.  A  plane  figure  whose  bound- 
ing line  is  everywhere  equally  dis- 
tant from  a  point  within,  called 
the  center,  is  a  Circle. 

1.  The  curved  line  that  bounds  a  circle  is  the  Circumfer- 
ence ^ 

2.  A  straight  line  passing  from  one  side  of  a  circle  to  the 
other,  through  the  center,  is  a  Diameter  ;  as  AB. 

3.  A  line  from  the  center  of  a  circle  to  the  circumference 
is  called  a  Radius.     It  is  lialf  a  diameter ;  as  CD  or  CA. 

Queries. — 1.  Can  there  be  a  center  with- 
out a  circumference? 

2.  What  is  the  surface  between  the  center 
and  circumference  called? 

1^°  Divide  the  diameter  of  a  circle  into 
10  equal  parts,  and  step  dividers  or  compasses 
around  the  circumference,  making  each  step 
equal  to  one  of  the  divisions  of  the  diameter. 
There  will  be  a  little  more  than  31  steps. 
Hence  the  circumference  is  a  little  more  than 
3.1  times  the  diameter.  In  geometry  it  is  shown  to  be  3.1416  times  the 
diameter. 

318.  Peinciple. — The    circumfereyice    of  any  circle    is 
3,14.16  times  its  diameter. 

What  is  the  circumference  of 

1.  A  circle  whose  diameter  is  1  ft.  ?     60  ft.  ?     27  in.  ? 

2.  A  circle  whose  radius  is  6  ft.  ?     12^  ft.  ?    2  ft.  6  in.  ? 
What  is  the  diameter  of  a  circle  whose 

3.  Radius  is  7i  in.?     13fin.?     19.08ft.? 


COMPOUND  NUMBERS. 


211 


4.  Circumference  is  6.2832  ft.  ?     12.5664  yd.  ?    1  ft. ? 

5.  What  is  the  radius  of  a  circular  field  whose  circum- 
ference is  320  rods  ? 

6.  What  is  the  circumference  of  a  6-inch  stove-pipe  ? 

319.  To  ti\ul  the  area  of  a 
circle. 

(a).  May  the  circle  AB  be  regarded 
as  made  up  of  a  vast  number  of  very 
small  triangles  ?  To  what  is  the  sum 
of  their  bases  equal  ?  Is  the  altitnde 
of  each  triangle  equal  to  the  radius  of 
the  circle  ? 

(5).  Is  the  area  of  all  the  triangles 
equal  to  the  area  of  the  circle  ? 

(c).  Since  the  area  of  a  triangle  is  found  by  multiplying 
its  base  by  half  its  altitude,  how  may  the  area  of  a  circle  be 
found  ? 

320.  Principle. — TJie  area  of  a  circle  is  expressed  ly 
the  product  of  the  numbers  that  represent  its  'circumference 
and  half  its  radius. 

This  principle  may  be  illustrated  as  follows  in  (a)  and  {h) : 

(a).  In  the  figure  ahc,  the  base  ab  is 
a  curved  line — an  arc  of  a  circle  whose 
center  is  c.  If  this  arc  be  pressed  up 
or  stretched  until  it  becomes  a 
straight  line,  the  sides  ac  and  be  will 
be  forced  farther  apart,  and  the  figure 
will  appear  SiS,  dec,  the  line  de  being 
equal  to  the  arc  ab,  and  the  other 
sides  remaining  unchanged..  But  co, 
the  radius,  is  a  little  longer  than  the  altitude  ck.  However,  by  increas- 
ing the  number  of  radii  the  arc  may  be  made  as  small  as  we  please;  and, 
as  the  arc  is  thus  made  smaller,  the  difference  between  the  radius  and 
the  altitude  becomes  continually  less.  When  the  arc  is  extremely  small, 
the  radius  and  the  altitude  are  regarded  as  equal. 


212 


SCHOOL  ARITHMETIC. 


(b).  Take  a  rubber-tired  wheel  with 
spokes,  as  WL.  Cut  it  through  and 
straighten  out  as  in  the  figure.  The  result- 
ing figures  1,  2,  3,  etc.,  are  nearly  triangles. 
The  sura  of  all  the  bases  is  easily  se^n  to  be 
the  circumference  of  the  wheel,  or  circle. 
The  area  of  all  the  triangles  is  equal  to  that 
of  the  circle,  and  their  common  altitude  is 
the  radius.     (See  Art.  319.) 

(c).  Regarding  the  circle,  then,  as  made 
up  of  many  triangles,  we   find  its  area  by 
multiplying  the  su7n  of  all  the  bases  (which 
is  the  circumference)  by  one-half  the 
common  altitude  (i.  e.,  i  the  radius). 
It  is  proved  in  geometry  that  this  is  the 
exact  area. 

Find  the  area  of  the  following     /»\/2\/3\/4\/5\/6\/7\/8\ 
circles  : 

1.  Radius  10,  circumference  63.833. 

62.832  X  5  =  314.16.     Or, 
iof  62.832  X  10  =  314.16. 

2.  Diameter  10  in.,  circumference  31.416  in. 

3.  Diameter  15  ft.,  circumference  47.124  ft. 

4.  Diameter  7.9578  ft.,  circumference  25  ft. 

5.  Diameter  15  yd.  7.  Circumference  100  ft. 

6.  Radius  13.5  rd.  8.  Circumference  1.5708  ft. 

9.  At  $50  an  acre,  what  is  the  value  of  a  circular  field 
whose  diameter  is  100  rods  ? 

10.  A  horse  is  tied  to  an  iron  weight  by  a  rope  20  feet 
long.  Upon  how  many  sq.  ft.  can  he  graze  ?  Draw  the 
figure. 

11.  The  fence  around  a  circular  field  is  500  rods  long. 
How  many  acres  in  the  field  ? 

For  convenience,  3.1416  is  usually  represented  by  the  Greek  letter  7t 
{pi),  the  diameter  by  d,  and  the  radius  by  r.  Then  the  circumference 
is  expressed  by  nd.     But  since  d  =  2r,  the  circumference  equals  27Cr ; 


Compound  numbers.  213 

and  the  area  equals  2'Kr  x  \r,or  it  x  r  x  r,  or  nr^.    Thus,  if  the  radius 
of  a  circle  is  3  ft.,  the  area  is  3.141G  x  3^  x  1  sq.  ft.  =  28.2744  sq.  ft. 

12.  Solve  examples  2,  5,  and  6  by  this  formula,  and  com- 
pare results  with  those  already  obtained. 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

321.  1.  The  perimeter  of  a  rectangular  field  is  200  rods, 
and  three  times  the  width  is  twice  the  length.  Find  the  area 
in  acres. 

2.  Each  of  the  three  sides  of  a  field  is  255J  feet  long.  The 
field  is  surrounded  by  a  fence  5  boards  high,  the  posts  being 
8  feet  apart.  If  the  posts  cost  35  cents  each,  and  the  boards 
2  cents  a  linear  foot,  what  did  the  posts  and  boards  cost  ? 

3.  The  distance  around  a  circular  lot  is  1000  feet.  If  I  plow 
around  the  lot  until  |  of  it  is  plowed,  how  many  square  feet 
will  remain  to  be  plowed  ? 

4.  Find  the  area  of  a  circular  ring  4  feet  wide,  the  radius 
of  the  outer  circle  being  32  feet. 

5.  Find  the  length  of  the  minute-hand  of  a  clock  whose 
point  moves  5  inches  in  15  minutes. 

6.  A  takes  924  steps  in  walking  around  a  field  40  rods 
long.  If  there  are  5  feet  in  two  of  his  steps,  how  many 
acres  in  the  field  ? 

7.  A  carpenter  had  a  plank  20  in.  wide,  from  which  he 
wished  to  saw  off  10  sq.  ft.  What  will  be  the  length  of  the 
piece  sawed  off  ? 

8.  I  have  a  rectangular  farm  whose  perimeter  is  240  rods. 
It  is  twice  as  long  as  it  is  wide.  How  many  acres  does  it 
contain  ? 

9.  A  map  is  4  sq.  ft.  4  sq.  in.  in  area,  and  is  drawn  on  a 
scale  of  1  inch  to  a  mile.  How  many  acres  are  represented 
on  the  map  ? 

10.  A  square  lawn  is  bordered  by  a  gravel  drive  10  yards 
wide.  The  drive  covers  4000  sq.  yd.  How  many  sq.  yd.  in 
the  enclosed  lawn  ? 


214  SCHOOL  ARITHMETIC. 

11.  If  ^  feet  of  fence  cost  $24.50,  what  will  be  the  cost  of 
fencing  a  square  field,  one  side  of  which  is  3b  feet  long  ? 

12.  What  is  the  area  of  a'cirfcle  whose  radius  is  /*  feet  ? 

13.  How  many  sq.  ft.  in  a  table  which  is  a  inches  long  and 
b  inches  wide  ? 

14.  The  distance  around  a  rectangular  field  is  p  rods.  If 
the  field  is  q  rods  long,  how  wide  is  it  ? 

15.  My  storeroom  is  75  ft.  long,  35  ft.  wide,  and  20  ft. 
higli,  the  vertex  of  the  roof  being  15  ft.  above  the  upper 
floor.  Making  no  allowance  for  doors  and  windows,  what 
will  be  the  cost  of  painting  at  $.18  a  sq.  yd.? 

MEASURES    OF    VOLUME. 

322.  Any  limited  portion  of  space  is  called  a  Solid.  A 
solid  has  three  dimensions^ — length,  breadth,  and  thickness. 

323.  A  solid  that  has  six  rectangular  sides,  or  faces,  is 
called  a  Rectangular  Solid. 

.1.  A  rectangular  side  or  surface  is  one  that  has  the  form 
of  a  rectangle. 

2.  Is  a  brick  a  rectangular  solid  ?  Wliy  ?  Name  three 
bodies  that  are  rectangular  solids. 

324.  A  regular  solid  with  six  square  faces  is 
called  a  Cube. 

1.  A  cube  has  twelve  edges.     Where  are  they  ? 

325.  A  Cubic  Inch  is  a  cube  whose  faces 
are  each  an  inch  long  and  an  incli  wide. 

1.  What  is  a  cubic  foot?  A  cubic  yard?  A  cubic  unit  ? 
A  solid  unit  ? 

326.  The  Volume  of  any  solid  or  body  is  the  number  of 
solid  or  cubic  units  it  contains. 

The  term  solid  contents  is  often  used  instead  of  volume. 


COMPOUND  NUMBERS. 


215 


A  solid  4  ft.  long,  3  ft.  wide, 
jind  5  ft.  high  (or  thick),  may 
be  divided  into  5  slabs,  each 
containing  3  rows  of  blocks,  as 
ABCD. 

Each  row  contains  4  blocks, 
OY  cubic  feet,  hence  3  rows  con- 
tain 12  cubic  feet. 

Since  there  are  12  cubic  feet 
in  1  slab,  in  5  slabs  there  are 
5  times  12  cubic  feet,  or  60 
cubic  feet,  which  is  the  vol- 
ume. 

(4  X  3  X  5)  X  1  cu.  ft.  =  60 
cu.  ft.  in  the  solid.  What  is 
the  unit  of  measure  here  ? 

327.  Principle.— ^Ae  vol- 
ume of  a  rectangular  solid  is  expressed  hj  the  product  of 
the  7iumhers  that  represent  its  length,  breadth,  and  thickness. 

Since  the  area  of  one  face  or  side  of  a  rectangular  solid  4 
ft.  long  and  3  ft.  wide  is  12  sq.  ft.,  it  will  be  observed  that 
the  number  of  cubic  feet  in  1  foot  of  thickness  is  the  same 
as  the  area  of  one  face.* 

Hence,  to  find  the  volume  of  a  rectangular  solid. 

Rule. — Multiply  the  mimber  of  cubic  units  in  one  tmit  of 
thickness  or  height  by  the  number  that  represents  the  height. 

All  dimensions  must  be  expressed  in  like  units. 

1.  An  iron  bar  is  12  inches  long,  1  inch  wide,  and  1  inch 
thick.     How:  many  cubic  inches  does  it  contain  ? 

2.  A  marble  slab  is  12  inches  square  and  1  inch  thick. 
How  many  cubic  inches  does  it  contain  ? 

3.  A  block  of  coal  is  12  inches  long,  12  inches  wide,  and 
12  inches  thick.  How  many  cubic  inches  in  it  ?  What  may 
this  solid  be  called  ?    Why  ? 


216  SCHOOL  ARITHMETIC. 

4.  A  cube  of  stone  measures  3  feet  on  each  side.  How 
many  cubic  feet  does  it  contain  ?  What  may  this  cube  be 
called  ?    Why  ? 

CUBIC   MEASURE. 

328.  Cubic  Measure  is  used  in  measuring  solids. 

Table. 

1728  cubic  inches  (cu.  in.)  =1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.), 

cu.  yd.     cu.  ft.      cu.  in. 
1        1^    27    =  46656 

1.  A  pile  of  wood  8  ft.  long,  4  ft.  wide,  and  4  ft.  high  is  called  a  cord. 
It  contains  128  cu.  ft. 

2.  In  measuring  masonry  or  stone,  24£  cu.  ft.  =  1  perch. 

3.  In  finding  the  length  of  walls,  masons  measure  the  outside.  The 
corners  are  thus  counted  twice. 

4.  A  cubic  foot  of  distilled  water  at  its  greatest  density  weighs  1000 
oz.  avoirdupois. 

WRITTEN    EXERCISES. 

329.  Reduce  : 

1.  3  cu.  ft.  to  cu.  in.  4.  2  A.  to  sq.  yd. 

2.  5  cu.  yd.  to  cu.  in.        5.  1  cu.  yd.  1  cu.  ft.  to  cu.  in. 

3.  6  cu.  yd.  to  cu.  ft.         6.  108  cu*.  ft.  to  cu.  yd. 

7.  Reduce  594  cu.  ft.  to  cubic  yards. 

8.  Reduce  50000  cu.  in.  to  higher  denominations. 

9.  What  part  of  a  cubic  yard  is  -^-^  of  a  cubic  foot  ? 

10.  Reduce  ^  cu.  yd.  to  lower  denominations. 

11.  How  many  cu.  in.  in  .75  of  a  cubic  foot  ? 

12.  A  log  2  feet  square  is  12  ft.  5  in.  long.  How  many 
cubic  inches  does  it  contain  ? 

13.  A  cellar  is  30  ft.  long,  24  ft.  wide,  and  8  ft.  deep. 
How  many  cu.  yd.  of  earth  were  removed  ? 

14.  How  many  cu.  ft.  of  air  in  a  room  16  ft.  square  and 
12  ft.  high  ? 


COxMPOUND  NUMBERS.  217 

15.  My  granary  is  8  ft.  long,  4  ft.  3  in.  wide,  and  5  ft. 
deep.     How  many  bushels  of  wheat  will  it  hold  ? 

16.  A  cubical  cistern  8  ft.  deep  contains  2  ft.  of  water. 
How  many  gallons  in  it  ? 

17.  Some  gold  miners  sunk  a  shaft  390  feet.  If  it  was  4 
ft.  by  8  ft.,  how  many  cu.  yd.  of  material  were  removed  ? 

18.  A  trough  3  ft.  wide  and  1  ft.  3  in.  deep  is  8  ft.  long. 
If  a  cu.  ft.  of  water  weighs  1000  oz.,  how  many  pounds  of 
water  will  fill  the  trough  ? 

19.  If  a  box  car  is  40  ft.  long,  8  ft.  wide,  and  9  ft.  high, 
how  many  blocks  of  marble,  each  4  ft.  by  2  ft.  by  1  ft.  G  in., 
will  it  hold  ? 

20.  A  two-quart  pail  was  half  full  of  water.  Several  frogs 
jumped  into  the  pail,  and  then  it  was  full.  What  was  the 
volume  of  the  frogs  ? 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

330.  1.  At  30  cents  a  bushel,  find  the  cost  of  a  box  of 
oats,  the  box  being  8  feet  long,  4  feet  wide,  and  4  feet  deep. 

2.  A  rectangular  trough  is  f  full  of  water.  After  35 
gallons  are  taken  out,  it  is  f  full.  What  is  the  depth  of  the 
trough,  the  length  being  10  feet  and  the  width  2J  ft.  ? 

3.  A  two-acre  pond  is  covered  with  ice  6  in.  thick.  If  a 
cu.  ft.  of  ice  weighs  896  ounces,  how  many  tons  of  ice  are  on 
the  pond  ? 

4.  A  tank  8  ft.  long,  IJ  ft.  wide,  and  8  in.  deep  will  con- 
tain how  many  pounds  of  water,  a  cu.  ft.  of  water  weighing 
62^  lb.  ? 

6.  What  is  the  difference  between  a  3-foot  cube  and  3 
cubic  feet  ? 

6.  A  cistern  6  ft.  long  and  4^  ft.  wide  holds  108  cu.  ft.  of 
water.  How  many  cu.  in.  of  zinc  will  be  required  to  line  the 
sides  and  bottom,  the  zinc  being  ^  in.  thick  ? 

7.  What  are  the  dimensions  of  a  rectangular  box  whose 


218  SCHOOL  ARITHMETIC. 

capacity  is  50274  cu.  ft.,   the   length,   breadth,   and   depth 
being  to  each  other  as  3,  2,  and  1  ? 

8.  A  contractor  received  S45.50  for  digging  a  cellar  18  ft. 
long  and  15  ft.  wide,  at  $.65  a  cu.  yd.  To  what  depth  did 
he  dig  it  ? 

9.  A  man  wishes  to  make  a  bin  to  contain  725  bushels,  the 
width  and  depth  to  be  equal,  and  the  length  to  be  double  the 
widtli.     What  must  be  its  dimensions  ? 

10.  How  thin  is  a  cu.  in.  of  gold  beaten  so  as  to  cover  a 
space  46  ft.  10  in.  by  41  ft.  8  in.? 

11.  How  many  gallons  of  water  can  be  poured  into  a 
bushel  measure  ? 

12.  A  vessel  3  inches  square  contains  some  water.  A  gold 
chain  dropped  into  the  water  raises  the  fluid  ^  in.  What  is 
the  volume  of  the  chain  ?  What  is  its  weight  if  the  gold 
weighs  19.2  times  as  much  as  water  and  a  cu.  ft.  of  the 
water  weighs  1000  oz.  ? 

SURVEYORS'  MEASURE. 

331.  In  measuring  boundaries  of  land,  in  locating  rail- 
roads, etc.,  and  in  computing  the  area  of  land,  surveyors  and 
engineers  make  use  of  measures  not  given  in  the  ordinary 
tables  under  '^Measures  of  Extension."  They  are  as  fol- 
lows : 

•  SURVEYORS'  LINEAR    MEASURE. 

Table. 

7.92  inches  =  1  link  (1.). 
25  links     =  1  rod  (rd.). 
.4  rods      =  1  chain  (ch.). 
80  chains  =  1  mile  (mi.). 

The  unit  of  surveyors'  measure  is  Gu7iter's  Chain,  which 
is  100  links  long. 


COMPOUND  NUMBERS.  219 

SURVEYORS"  SQUARE  MEASURE. 

Table. 

16  square  rods      =  1  square  chain  (sq.  ch.). 
10  square  chains  =  1  acre  (A.). 
640  acres  =  1  square  mile  (sq.  mi.). 


1  mile  square  =  1  section. 
36  sections         =  1  township. 

MEASURES    OF    TIME. 

332.  The  time  required  for  the  earth  to  rotate  once  on 
its  axis  is  called  a  day,  which  is  the  unit  of  time.  Other 
divisions  of  time  are  shown  in  the  following 

Table. 

60  seconds  (sec.)  =  1  minute  (min.). 
60  minutes  =  1  hour  (hr.). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk). 

365  days  =  1  year  (yr.). 

366  days        =  1  leap  year  (yr.). 

da.  hr.   min.    sec. 
1  =  24  =  1440  =  86400. 

1.  A  period  of  100  years  is  called  a  century,  10  years  a 
decade. 

2.  In  business  transactions  30  days  are  usually  considered 
a  month. 

3.  The  time  from  midnight  to  noon  is  caW^di  forenoon — 
A.M.  ;    that  from  noon  to  midnight  is  called   afternoon — 

P.M. 

THE     CALENDAR. 

333.  The  system  of  reckoning  time  by  years  and  months, 
as  found  in  our  almanacs,  is  called  a  Calendar. 


2^0  SCHOOL  ARITHMETIC. 

1.  The  time  of  one  fevbhition  of  the  earth  around  the  sun^— the  solar 
year^-^is  365  da.  5  hr.  48  rain.  46  sec,  nearly.  For  ordinary  purposes  it 
is  impol^tatit  that  the  year  contain  ah  exact  number  of  days.  The  pres- 
ent calendar  year  secures  this  result.  Its  length  is  sometimes  365  and 
Sometimes  866  days,  but  its  average  length  is  almost  exactly  equal  to 
that  of  the  solar  year.  The  year  of  366  days  is  called  a  leap-year,  and 
those  years  are  leap-years  whose  date  numbers  are  exactly  divisible  hy 
4,  except  centen7iial  years,  whose  dates  must  he  divisible  by  4OO. 

2.  On  the  supposition  that  the  length  of  the  year  was  365^  days,  Julius 
Caesar  introduced  a  calendar  in  which  every  year  whose  date  number  is 
exactly  divisible  by  4  was  to  consist  of  366  days,  and  all  other  years  of 
365.  But  as  the  year  does  not  contain  exactly  365^  days,  Caesar's  year 
was  thus  11  minutes  and  about  14  seconds  too  long,  and  this  would  ac- 
cumulate in  400  years  to  a  little  over  3  days.  In  1582,  Pope  Gregory 
XIII. ,  aiming  to  correct  this  error,  arranged  that  only  each  fourth  cen- 
tennial year  {those  exactly  divisible  by  40O)  should  be  a  leap-year.  The 
Gregorian  Calendar  still  leaves  a  very  slight  error,  which  will  not  amount 
to  a  day  until  about  a.d.  5200. 

3.  The  calendar  of  Caesar  is  known  as  the  Julian  Caleyidar,  or  Old 
Style  (0.  S.)  ;  that  which  Gregory  substituted  is  called  the  Gregorian 
Calendar,  or  New  Style  (N.  S.).  In  changing  from  one  calendar  to  the 
other,  Gregory  dropped  out  10  days,  so  that  the  day  after  Oct.  4,  1582, 
was  called  Oct.  15.  England  adopted  the  New  Style  in  1750,  by  which 
time  it  had  become  necessary  to  drop  out  11  days.  The  difference  in 
1899  was  12  days.     From  1900  to  2100  it  will  be  13  days. 

Example. — Columbus  discovered  America,  October  12, 
1492,  0.  S.     What  is  the  date  N.  S.  ? 

WRITTEN    EXERCISES. 

334.  Reduce  to  lower  denominations  : 

1.  3  hr.  10  min.  15  sec.  4.  1  wk.  1  da.  1  hr. 

2.  2  da.  2  hr.  30  min.  5.  3  sq.  yd.  5  sq.  ft. 

3.  1  wk.  7  hr.  25  min.  6.  5  cwt.  87  lb. 
Reduce  to  higher  denominations  : 

7.  1000  sec.  9.  5050  hr.  11.  3675  da. 

8.  2345  min.  10.  13258  wk.  12.  3636  cu.  in. 

13.  What  part  of  a  day  is  |  of  an  hour  ? 


COMPOUND  NUMBERS.  221 

14.  Reduce  f  of  a  day  to  hours  and  minutes. 

15.  AVhat  decimal  part  of  a  day  is  8  lir.  40  min.  ? 

16.  In  1894  how  many  days  from  January  1st  to  March 
15  ?    In  1896  ?     In  1900  ? 

17.  How  many  decades  in  one  century  and  15  years  ? 

18.  John  was  paid  $2.10  for  working  from  7  a.m.  till  15 
minutes  past  12  m.     How  much  was  that  an  hour  ? 

19.  George  worked  5  da.  6  hr.  30  min.  at  the  rate  of  $2  a 
day.     How  much  did  he  earn  ? 


CIRCULAR    OR    ANGULAR    MEASURE. 

335.  An  Arc  of  a  circle  is  any  part 
of  the  circumference ;  as  EB  or  BD. 

336.  When  two  lines  are  drawn 
from  the  circumference  to  the  center, 
the  arc  between  them  is  the  measure 
of  the  angle  formed. 

Thus,  the  arc  DE  is  tlie  measure  of  the 
angle  DCE. 

337.  In  measuring  angles,  the  circumference  is  divided 
into  360  equal  i)arts,  called  degrees  j  each  degree  into  60 
parts,  called  minutes  j  and  each  minute  into  60  parts,  called 
seconds. 

1.  The  length  of  a  degree  varies.  A  degree  is  3^0  of  a  circumference, 
and  the  greater  the  circumference  the  greater  the  degree,  or  arc. 

2.  The  angle  measured  by  a  degree  does  not  vary. 

338.  Circular  or  Angular  Measure  is  used  in  measur- 
ing angles,  arcs  of  circles,  in  determining  latitude  and 
longitude,  the  location  of  vessels  at  sea,  etc. 

Table. 

60  seconds  (")  =  1  minute  ('). 
60  minutes       =  1  degree  (°). 
360  degrees        =  1  circumference  (cir.). 


222  SCHOOL  ARITHMETIC. 

One  fourth  of  a  circumference,  or  90°,  is  called  a  quadrant. 
A  minute  of  the  circumference  of  the  earth  is  called  a  geo- 
graphical mile. 

Miscellaneous  Tables. 

Paper. 


12  things  =  1  dozen. 
12  dozen  =  1  gross. 
12  gross    =  I  great  gross. 

2  things  =  1  pair. 

6  things  =  1  set. 
20  things  =  1  score. 


24  sheets     =  1  quire. 
20  quires     =  1  ream. 

2  reams     =  1  bundle. 

5  bundles  —  1  bale. 


ORAL     EXERCISES. 

339.  1.  Mrs.  A  has  30  knives.     How  many  sets  has  she  ? 

2.  Mary  sold  5  doz.  eggs  at  2  cents  apiece.  How  much  did 
ghe  get  for  them  ? 

3.  Harry  bought  36  lemons  at  200  a  dozen.  What  did 
they  cost  him  ? 

4.  Sarah  bought  5  quires  of  paper  at  15  cents  a  quire,  and 
sold  it  at  a  cent  a  sheet.     How  much  did  she  gain  ? 

5.  Ten  years  ago  Mr.  Smith  was  3  score  and  10  years  of 
age.     How  old  is  he  now  ? 

6.  Half  a  dozen  is  what  part  of  a  gross  ? 

7.  What  is  the  ratio  of  a  dozen  to  a  score  ? 

8.  How  often  is  %\  contained  in  1.75  ? 

9.  A  lady  sold  200  eggs  at  20  cents  a  dozen,  and  took  her 
pay  in  sugar  at  6  cents  a  pound.  How  many  pounds  did  she 
get? 

10.  What  has  the  same  ratio  to  a  foot  that  a  yard  has  to  an 
inch  ? 

ADDITION. 

340.  In  simple  numbers  the  scale  is  uniform,  10  units  of 
any  order  being  equal  to  1  of  the  next  higher  order.  In 
compound  numbers  the  scale  is  varying. 

In  U.  S.  money  the  scale  is  uniform,  and  it  is  to  be  observed  that 


COMPOUND  NUMBERS.  223 

while  $2.55  is  a  simple  number,  yet  when  written  in  the  separate  units 

$2  5(1.   5^,  it  may  be  regarded  as  a  compound  number. 

The  addition,  subtraction,  multiplication,  and  division  of  compound 
numbers  differ  little  in  theory  from  the  like  operations  in  simple  num- 
bers. Their  varying  scale,  however,  causes  the  numbers  to  be  written 
somewhat  differently,  and  makes  a  slight  difference  in  the  processes. 

WRITTEN     EXERCISES. 

1.  What  is  the  sum  of  $7  2d.   4^'  5m.,   $3  8d.    0^    7m., 


and  $2  7d.  5^  Im.  ? 

M 

{b) 

$  d.   0   m. 

7  2   4   5 

$7,245 

3  8   6   7 

3.867 

2  7   5   1 

2.751 

13    8      6      3  1^13.863 

The  given  numbers  may  be  written  either  as  in  (a)  or  (&),  since  they 
have  a  decimal  scale.  Units  of  the  same  order  should  stand  in  the 
same  column.     The  numbers  in  (a)  are  added  as  follows  : 

The  sum  of  the  mills  is  13,  which  is  equal  to  1  cent  and  3  mills.  We 
write  the  8  in  the  column  of  mills,  and  add  the  1  to  the  cents,  making 
160,  or  1  dime  and  60.  Writing  the  6  in  the  column  of  cents,  we  add 
the  Id.  to  the  dimes,  making  18d.,  or  $1  and  8d.  We  write  the  8  in 
the  column  of  dimes,  and  add  the  $1  to  the  dollars,  making  $13. 

I^^Have  the  pupil  add  the  numbers  in  {h),  and  compare  the  process 
with  that  just  explained. 

2.  What    is  the   sum   of  16s.  9d.   1  far.,  18s.  8d.  3  far., 
3  far.  ? 

We  write  units  of  the  same  order  in  the  same 
vertical  column,  regardless  of  the  number  of  places 
they  may  occupy.  Thus,  lOd.,  being  less  than  1 
shilling,  must  be  placed  in  the  pence  column. 

The   sum  of  the  farthings  is  7,  or  Id.  8   far. 
2      K     A       '^  Writing  the  8  in  the  column  of  farthings,  we  add 

the  Id.  to  the  pence,  making  28d.,  or  2s.  4d.     Writ- 
ing the  4  in  the  column  of  pence,  we  add  the  2s.  to  the  shillings,  making 
45s.,  or  £2  5s.,  which  we  write  in  the  proper  columns. 
Query. — Why  do  we  not  *'  carry  "  one  for  every  tejh  ? 


and  9s 

.  lOd.  { 

£ 

s. 

d. 

far. 

16 

9 

1 

18 

8 

3 

9 

10 

3 

224  SCHOOL  ARITHMETIC. 

3.  Find  the  sum  of  3  gal.  2  qt.  1  pt.,  7  gal.  3  qt.,  and  8 
gal.  1  qt.   1  pt. 

4.  Find  the  sum  of  3  bu.  1  pk.  7  qt.,  9  bu.  3  pk.  5  qt. 
1  pt.,  12  bu.  2  pk.,  and  7  bu.  1  pk.  6  qt. 

5.  Add  8  lb.  5  oz.  12  pwt.  16  gr.,  12  lb.  9  oz.  4  pwt.  9  gr., 
15  lb.  11  oz.  19  pwt.  22  gr.,  and  10  oz.  17  pwt. 

6.  What  is  the  sum  of  2  T.  15  cwt.  40  lb.,  5  T.  8  cwt. 
75  lb.,  4  T.  9  cwt.  85  lb.,  and  13  T.  3  cwt.  ? 

7.  Add  1  mi.  40  rd.  5  yd.  2  ft.  6  in.,  12  mi.  185  rd.  2  yd. 
1  ft.  8  in.,  5  mi.  316  rd.  4  yd.  7  in.,  8  mi.  1  yd.  2  ft.  3  in., 
and.  18  rd.  4  yd.  10  in. 

When  a  fraction  occurs  in  the  sum,  it  should  be  reduced  to  lower  de- 
nominations, and  added  to  the  proper  columns. 

8.  What  is  the  sum  of  80  sq.  rd.  25  sq.  yd.  5  sq.  ft.  75  sq. 
in.,  136  sq.  r-d.  12  sq.  yd.  8  sq.  ft.  120  sq.  in.,  48  sq.  rd.  9  sq. 
yd.  3  sq.  ft.  56  sq.  in.,  and  108  sq.  rd.  136  sq.  in.  ? 

9.  Find  the  sum  of  18  cu.  yd.  13  cu.  ft.  87  cu.  in.,  9  cu. 
yd.  21  cu.  ft.  1236  cu.  in.,  4  cu.  yd.  25  cu.  ft.  1600  cu.  in., 
and  18  cu.  ft.  760  cu.  in. 

10.  Add  12  hr.  40  min.  34  sec,  7  hr.  8  min.  50  sec,  18 
hr.  25  min.  26  sec,  19  hr.  15  min.  45  sec,  and  6  hr.  50  min. 
12  sec 

11.  What  is  the  sum  of  £16  12s.  7d.  3  far.,  £4  18s.  lid. 
1  far.,  £2  16s.  9d.  2  far.,  and  £3  8s.  7d.  1  far. 

12.  Find  the  value  of  f  bu.  +  |  pk.  +  5|  qt. 

I  bu.  =3  pk.  4  qt. 
i  pk.  =  6 

H  qt.  =  5         1  pt. 


I  bu.  +  f  pk.  +  5i  qt.  =  3  pk.  7  qt.  1  pt. 

13.  What  is  the  value  of  ^^  wk.  +  f  da.  +  f  hr.  r 

14.  Add  3i  lb.  8  oz.  13^  pwt.,  li  lb.  3^  oz.  10  pwt.  18  gr., 
and  7  lb.  6  oz.  17  pwt.  15  gr. 

15.  Add  45  gal.,  3.7  qt.,  and  1.5  pt. 
Solution  same  as  in  example  13. 


COMPOUND  NUMBERS.  225 

16.  Find  the  sum  of  .25  mi.,  1.15  mi.,  and  120  rd.  18  ft. 

17.  The  latitude  of  Pittsburg  is  40°  27'  36"  north,  and 
that  of  the  Cape  of  Good  Hope  is  33°  56'  3"  south.  How 
many  degrees  between  the  two  places  ? 

Why  do  we  add  to  find  the  diflference  in  the  latitude  of  these  places  ? 
Draw  figure  to  illustrate. 

18.  A  farmer  sold  corn  as  follows  :  To  A,  75  bu.  1  pk.;  to 
B,  37  bu.  3  pk.;  to  0,  110^  bu.;  to  D,  18  bu.  3i  pk. ;  to  E, 
42^  bu.     How  much  did  he  sell  to  all  ? 

SUBTRACTION. 

WRITTEN    EXERCISES. 

341.  1.  Mr.  A  had  $20  8d.  5^,  and  spent  $12  9d.  7^. 
How  much  had  he  left  ? 

(a)  (b) 

$    d.    0 

20     8     5  $20.85 

12     9     7  12.97 


7     8     8  $7.88  • 

The  given  numbers  may  be  written  either  as  in  (a)  or  (b),  units  of  the 
same  order  standing  in  the  same  column.  The  numbers  in  (a)  are  sub- 
tracted as  follows  : 

We  can  not  take  7^  from  5^,  hence  we  take  Id.  (10^)  from  the  8d., 
and  add  it  to  the  5^,  making  15^.  Then  7^  from  150  leaves  80,  which 
we  write  in  the  column  of  cents. 

Having  taken  Id.  from  the  8d.,  only  7d.  remains,  from  which  9d.  can- 
not be  taken.  Hence  we  take  $1  (lOd.)  from  the  $20,  and  add  it  to  the 
7d.,  making  17d.  Then  9d.  from  17d.  leaves  8d.,  which  we  write  under 
dimes. 

Having  taken  $1  from  the  $20,  only  $19  remain.  Then  $12  from  $19 
leaves  $7,  which  we  write  as  the  dollars  of  the  remainder. 

m^"  Have  the  pupil  subtract  the  numbers  in  (&),  and  compare  the 
process  with  that  just  explained. 

2.  B  has  £10  8s.  6d.  and  D  has  £6  5s.  lOd.  less  than  B. 
How  much  has  D  ? 
15 


226  SCHOOL  ARITHMETIC. 

3.  From  5  bu.  1  pk.  6  qt.  1  pt.,  take  2  bu.  3  pk.  5  qt. 

4.  From  13  gal.  3  qt.,  take  8  gal.  2  qt.  1  pt. 

.    6.  Take  2  T.  18  cwt.  90  lb.  from  4  T.  15  cwt.  80  lb.  ' 

6.  From  3  cu.  yd.  take  21  cu.  ft.  1628  cu.  in. 

7.  Mr.  B  had  a  two-acre  lot  from  which  he  sold  1  A.  75 
sq.  rd.     How  much  had  he  left  ? 

8.  From  a  barrel  containing  41  gal.  2  qt.  1  pt.  of  molasses, 
10  gal.  3  qt.  were  sold  at  one  time  and  23  gal.  1  qt.  1  pt.  at 
another  time.     How  much  remained  in  the  barrel  ? 

9.  From  10  lb.  10  oz.  10  pwt.  10  gr.,  take  5  lb.  11  oz. 
12  pwt.  13  gr. 

10.  A  man  ate  lunch  at  12  o'clock  15  min.  p.m.,  and 
dinner  at  6  o'clock  45  min.  p.m.  How  long  was  it  between 
those  meals  ? 

11.  The  latitude  of  Pittsburg  is  40°  27'  36"  north,  and 
that  of  Washington  is  38*=^  53'  39"  north.  Find  their  differ- 
ence of  latitude. 

12.  From  f  A.  take  42.5  sq.  rd. 

sq.  rd.      sq.  yd.     sq.  ft.       sq.  in. 
I  A.        =  100 
42.5  sq.  rd.  =    42  15  1  18 


|A.-442.5sq.  rd.  =    57  14i  7  126 

4     =     2  36 


"      =57  15  1  18 

Query. — Why  do  we  change  i  sq.  yd.  to  2  sq.  ft.  36  sq,  in.  ? 

13.  From  |  mi.  take  234|  rd. 

14.  From  f  wk.  take  2.6  da. 

15.  Take  1  pk.  1  qt.  1  pt.  from  .7  bu. 

16.  From  f  gross  take  6f  doz. 

17.  How  long  was  it  from  May  12,  1892,  to  July  4,  1899  ? 

The  later  date  expresses  the  greater  period  of  time, 
hence  it  is  the  minuend.     The  later  date  is  the  4th 
day  of  the  7th  month  of  1899.     The  other  date  is  the 
7     1     22        12th  day  of  the  5th  month  of  1892. 


yr. 

mo. 

day. 

1899 

7 

4" 

1892 

5 

12 

COMPOUND  NUMBERS.  227 

We  subtract  as  in  other  compound  numbers,  considering  30  days  as 
a  month,  as  given  in  the  table. 

Query. — Does  the  remainder  express  the  eocact  time  between  the  given 
dates  ?    Why  not  ? 

Jt;^  To  find  the  exact  time  between  two  dates  (as,  for  example,  Aug. 
24  and  Dec.  3,  same  year)  we  must  proceed  as  follows  : 
7  +  80  +  31  +  30  4-  3  =  101. 

That  is,  there  are  7  more  days  in  Aug.,  30  in  Sept.,  31  in  Oct.,  30  in 
Nov.,  and  3  in  Dec.  The  sum  is  the  difference  in  time  expressed  in 
days. 

18.  The  War  between  the  States  began  April  11,  1861,  and 
ended  April  9,  1865.     How  long  did  it  continue  ? 

19.  Find  the  time  from  Oct.  15,  1812,  to  June  3,  1912. 

20.  A  note  dated  June  19,  1897,  was  paid  Oct.  12,  1898. 
How  long  did  it  run  ? 

21.  What  is  your  age  to-day  ? 

22.  How  long  is  it  from  to-day  to  Feb.  29,  1920  ? 

23.  Independence  was  declared  July  4,  1776.  How  long 
since  that,  important  event  occurred  ? 

MULTIPLICATION. 

WRITTEN     EXERCISES. 

342.  1.  Multiply  4  gal.  3  qt.  1  pt.  by  9. 

(*) 
9  times  1  pt.  =  9  pt.,  or  4  qt.  1  pt. 
9  times  3  qt.  +  4  qt.  =  31  qt.,  or  7  gal.  3  qt. 
9  times  4  gal.  +  7  gal.  =  43  gal. 
43         3        1  .".  the  product  is  43  gal.  3  qt.  1  pt. 

The  calculation  is  conveniently  made  as  in  (a). 

2.  Multiply  £6  12s.  8d.  by  5. 

3.  Multiply  4  lb.  9  oz.  6  pwt.  13  gr.  by  11. 

4.  Multiply  6  wk.  4  da.  8  hr.  20  min.  by  12. 

5.  A  grocer  bought  8  bags  of  chestnuts,  each  containing 
3  bu.  1  pk.  5  qt.     How  many  did  he  buy  ? 


(«) 

gal. 

qt. 

pt. 

4 

3 

1 

9 

228  SCHOOL  ARITHMETIC. 

6.  If  he  sold  the  chestnuts  at  6  cents  a  qt.,  how  much  did 
he  get  for  them  ? 

7.  What  is  the  difference  in  time  between  366  common 
years  and  366  leap  years  ? 

8.  How  far  is  it  around  a  square  that  measures  4  yd.  2  ft. 
8  in.  on  a  side  ? 

9.  How  much  calomel  can  be  put  into  20  bottles,  if  eacli 
bottle  holds  3  6  3  2   gr.  xv  ? 

10.  If  $20  will  buy  IT.  6cwt.  801b.  of  hay,  how  much 
hay  will  $240  buy  ? 

11.  When  silver  is  worth  $.60  an  ounce,  how  much  must 
be  paid  for  5  oz.  12  pwt.  16  gr.  ? 

12.  When  vinegar  is  6  cents  a  quart,  how  much  must  be 
paid  for  J  gal.  ? 

DIVISION. 

WRITTEN    EXERCISES. 

343.  1.  A  bbl.  of  syrup  containing  43  gal.  3  qt.  1  pt. 
was  shared  equally  by  9  persons.  How  much  did  each  one 
get? 

(a)  (h) 

43  gal.  -1-9  =  4  gal,,  and  7  gal.  (28  qt.)  remaining. 
gal.     qt.     pt.      28  qt.    +3qt.=81qt. 
Q\±^       ^        1       31  qt.    -7-  9  =  3qt.,  and  4  qt.  (8  pt.)  remaining. 

^M '1 i       8  pt.    +1  pt.  =  9  pt. 

4        3       1        9pt.  -^9  =  lpt. 

.".  the  quotient  is  4  gal.  3  qt.  1  pt. 

Query. — Is  the  dividend  here  the  product  in  example  1  in  Art.  848  ? 
How  do  we  find  one  of  the  two  factors  when  the  product  and  the  other 
factor  are  given  ? 

2.  If  5  horses  eat  11  T.  16  cwt.  80  lb.  of  hay  in  a  year,  how 
much  does  one  horse  eat  ? 

3.  The  distance  around  a  square  field  is  155  rd.  5  yd.  2  ft. 
What  is  the  length  of  one  side  ? 


COMPOUND  NUMBERS.  229 

4.  If  25  bu.  3  pk.  of  oats  fill  9  bags  of  the  same  capacity, 
how  much  do  2  bags  contain  ? 

5.  The  weight  of  8  silver  chains  is  18  oz.    12  pwt.  16  gr. 
What  is  the  average  weight  ? 

6.  A   druggist   made    3  2    31  gr.  x    of    calomel   into   15 
powders.     How  much  was  in  each  powder  ? 

7.  The  distance  around  a  rectangular  field  is  128  rd.  12  ft., 
and  one  side  is  42  rd.  4  ft.     What  is  the  length  of  one  end  ? 

8.  How  many  barrels  will  contain  30  bu.  1  pk.   4  qt.  of 
chestnuts,  if  each  barrel  holds  3  bu.  1  qt.  ? 

Reduce  botli  numbers  to  the  same  denomiiifttion,  ami  divide, 

9.  If  one  man  can  earn  £4  128.  lOd.  in  a  week,  how  many 
men  can  earn  £46  8s.  4d.  in  the  same  time  ? 

10.  How  often  can  1  ft.  8  in.  be  sawed  from  a  board  13  ft. 
4  in.  long  ? 

11.  How  many  dress  patterns  of  14f  yd.  each  can  be  cut 
from  a  piece  of  cloth  containing  44^  yd.  ? 

12.  How  many  blocks  1  ft.  square  can  be  cut  from  a  board 
12  ft.  long  and  2  ft.  wide  ? 

13.  A  two-acre  lot  is  8  rd.  wide.      How  many  yards  is  it 
around  the  lot  ? 

14.  Six  men  and  2  boys  weigh  1152  lb.  12  oz.     If  each  boy 
weighs  70  lb.  10  oz.,  what  is  the  average  weight  of  a  man  ? 

15.  How  many  steel  rails,  each  24  ft.  long,  are  required 
for  a  mile  of  sing'le-track  railroad  ? 

16.  l^ow  often  can  a  bucket  that  holds  2  gal.  3  qt.  1  pt. 
be  filled  from  a  hhd.  containing  60  gal.  1  qt.  1  pt.  ? 

17.  If  each  step  measures  2  ft.  8  in.,  how  many  steps  will 
a  man  take  in  walking  |  of  a  mile  ? 

18.  A  field  is  165  ft.  long  and  66  ft.  wide.     What  will  it 
cost  to  fence  it  at  $3  a  rod  ? 

19.  If  I  exchange  $1023.10  for  English  money,  how  many 
pounds,  shillings,  and  pence  do  I  receive  ? 


REVIEW    WORK. 


ORAL    EXERCISES. 


344.  1.  What  is  the  difference  between  a  square  yard  and 
3  square  feet  ? 

2.  What  is  the  difference  between  a  cubic  yard  and  3  cubic 
feet? 

3.  What  will  a  gallon  of  cream  cost  at  12  cents  a  pint  ? 

4.  At  $4  a  bushel,  what  will  be  the  cost  of  3  pk.  4  qt.  of 
clover  seed  ? 

5.  How  many  inches  in  5  feet  9  in.  ? 

6.  In  3.5  pwt.  how  many  grains  ? 

7.  What  will  2  gallons  of  vinegar  cost,  if  2  pints  cost  8 
cents  ? 

8.  A  field  is  40  rods  long  and  half  as  wide.     How  many 
acres  in  it  ? 

9.  If  9  pints  of  chestnuts  cost  45  cents,  how  much  will 
half  a  bushel  cost  ? 

10.  How  many  ounces  in  6^  lb.  avoirdupois  ? 

11.  At  a  cent  a  pound,  how  much  will  a  ton  of  iron  cost  ? 

12.  How  many  square  feet  in  5  sides  of  a  4-inch  cube  ? 

13.  The   distance   around   a   square  surface  is  48  inches. 
How  many  square  feet  does  it  contain  ? 

14.  From  }  of  a  score  take  |  of  a  dozen. 

15.  If  you  sleep  8  hours  each  night,  how  many  days  do 
you  sleep  in  2  Aveeks  ? 

16.  Mr.  B  left  home  on  Friday,  and  was  gone  25  days. 
On  what  day  of  the  week  did  he  return  ? 

17.  My  horse   eats  9   qt.  of  oats  a  day.     How  long  will  2 
bu.  ^  pk.  last  him  ? 


REVIEW  WORK.  231 

18.  A  rope  is  10  yards  long.  Into  how  many  pieces  each 
1^  ft.  long  can  it  be  cut  ? 

19.  If  there  are  G  ties  to  every  rod  of  track,  how  many  ties 
will  it  take  for  ^  of  a  mile  of  double-track  raih'oad  ? 

20.  How  many  pens  in  ^  gross  and  1^  dozen  ? 

21.  How  many  square  inches  in  the  surface  of  a  rectangle 
12  feet  long  and  ^  inch  wide  ? 

22.  A  lady  put  a  half  gallon  of  perfume  into  gill  bottles. 
How  many  bottles  were  required  ? 

23.  Mr.  A's  horse  is  15  hands  high,  and  Mr.  B's  is  16 J^ 
hands  high.     How  many  inches  higher  is  Mr.  B's  horse  ? 

24.  How  many  cubic  yards  of  air  in  a  room  4  yards  wide, 
6  yards  long,  and  9  feet  high  ? 

25.  How  many  days  between  Feb.  10  and  March  10,  1900  ? 

26.  A  man  traveled  g  of  the  distance  around  a  circular 
park.     Through  how  many  degrees  did  he  travel  ? 

27.  If  I  receive  $^  for  working  45  minutes,  how  mucli 
should  I  receive  for  9  hours'  labor  ? 

28.  What  is  the  entire  surface  of  a  cube  whose  edge  is  a 
feet  ?     What  is  the  volume  ? 

29.  What  is  the  value  of  b  gallons  of  milk  at  a  cents  a  pint  ? 

30.  If  p  bales  of  hay  weigh  q  pounds,  how  much  will  a 
bales  weigh  ? 

WRITTEN     EXERCISES. 

345.  1.  A  lot  is  120  feet  deep  and  60  feet  front.  How 
many  square  yards  does  it  contain  ? 

2.  A  gardener  put  3  bu.  1  pk.  7  qt.  of  berries  into  quart 
boxes.     How  many  boxes  were  required  ? 

3.  How  many  quarts  of  milk  will  fill  a  peck  measure  ? 

4.  How  many  ounces  of  quinine  in  5  lb.  avoirdupois  ? 

6.  What  will  be  the  cost,  at  $.35  a  square  yard,  of  painting 
5  sides  of  a  cube  whose  edge  is  4  feet  ? 

6.  How  many  five-grain  pills  can  a  druggist  make  front 
35  ^l  of  calomel? 


232  SCHOOL  ARITHMETIC. 

7.  How  many  gold  dollars  weigh  a  pound  avoirdupois  ? 

8.  How  many  gold  dollars  can  be  coined  from  a  pound  of 
gold  ? 

9.  How  many  pounds  of  silver  are  required  to  coin  100 
silver  dollars  ? 

10.  If  a  boy  idles  away  ^  hr.  a  day,  how  much  time  will 
he  lose  in  a  year  ? 

11.  A  20-acre  field  is  80  rods  long.  Find  the  cost  of  fenc- 
ing it  at  $.75  a  rod. 

12.  The  wheel  of  a  bicycle  is  7  ft.  4  in.  in  circumference. 
How  often  will  it  revolve  in  going  a  mile  ? 

13.  At  2 J  cents  a  pound,  how  many  barrels  of  flour  can  be 
bought  for  $53.90? 

14.  The  diameter  of  a  circular  field  is  40  rods,  and  the  side 
of  a  square  field  is  the  same.  Which  has  the  greater  area, 
and  how  much  ? 

16.  Find  the  cost  of  fencing  the  fields  just  mentioned,  at 
$1.50  a  rod. 

16.  When  land  is  worth  $80  an  acre,  what  is  the  value  of  a 
field  in  the  form  of  a  trapezoid,  whose  altitude  is  30  rods, 
and  whose  parallel  sides  are  48  and  32  rods  respectively  ? 

17.  Bought  2  lb.  silver  by  avoirdupois  weight,  paying  $8  a 
pound,  and  sold  it  by  troy  weight  at  80  cents  an  ounce.  How 
much  did  I  gain  ? 

18.  Bought  3  lb.  quinine  at  $4.50  a  pound  avoirdupois, 
and  sold  it  at  $6  a  pound  apothecaries'  weight.  What  did 
I  gain  ? 

19.  A  traveler  returning  from  Europe  had  £10,  10  shil- 
lings, 10  francs,  and  10  marks,  which  he  exchanged  for  U.  S. 
money.     How  much  did  he  get  ? 

20.  A  man  in  Boston  bought  2880  lb.  of  wheat,  at  $.85  a 
bushel,  and  sold  it  at  $.91  a  bushel.  How  much  did  he 
gain  ? 

21.  The  altitude  of  a  triangular  field  is  32  rods,  and  its 
area  is  8  aeres.     Find  the  length  of  the  base. 


R?:VIEW   WOKK.  233 

22.  What  is  the  radius  of  awheel  if  an  arc  of  15°  of  its  cir- 
cumference is  1  ft.  2  in.  in  length  ? 

23.  Find  the  cost  of  8  yd.  1  ft.  5  in.  of  i)ipe,  4  lb.  to  the 
foot,  at  24^/'  a  pound. 

24.  What  is  the  value  of  ^  of  a  square  mile  of  land  at  $G 
an  acre  ? 

25.  How  many  steel  rails  32  feet  long  are  needed  to  build 
2  miles  of  double-track  railway  ? 

26.  Bought  6  gross  of  lead  pencils  at  $3.50,  and  sold  them 
at  5  cents  apiece.     Find  the  gain. 

27.  My  little  girl  is  ''  worth  her  weight  in  gold  "  dollars. 
If  she  weighs  30  pounds,  what  is  she  worth  ? 

28.  I  sold  138  eggs  at  30  cents  a  dozen,  and  enough  butter 
at  35  cents  a  pound  to  make  my  total  receipts  $4.50.  llow 
many  pounds  of  butter  did  I  sell  ? 

29.  The  water  in  my  cistern,  which  is  G  ft.  long,  5  ft.  wide, 
and  8  ft.  deep,  weighs  15000  pounds.  Is  the  cistern  full  ?  If 
so,  what  is  the  weight  of  a  cubic  foot  of  water  ? 

30.  Mr.  H  has  a  feed-box  8  ft.  long,  3  ft.  wide,  and  4  ft. 
high.  AVhen  it  is  half  full  of  oats,  how  many  bushels  does 
it  contain  ? 

31.  How  many  square  feet  of  zinc  will  be  required  to  line 
the  sides  and  bottom  of  a  bin  12  ft.  square  and  5  ft. 
high? 

32.  If  tobies  cost  $9.50  a  thousand,  and  a  man  smokes  4  a 
day,  what  is  the  amount  of  his  annual  toby  bill  ? 

33.  How  many  cubes  measuring  6  inches  each  way  can  be 
cut  from  a  cubic  yard  of  marble  ? 

34.  Measure  your  schoolroom,  and  find  how  many  cubic 
feet  of  air  there  are  to  each  pupil. 

35.  What  will  be  the  cost  of  painting  a  box  2  feet  square 
and  10  feet  higli,  at  $.25  a  sq.  yd.? 

36.  How  many  square  yards  of  canvas  will  be  needed  to 
cover  two  boxes,  each  of  whose  bases  is  6  feet  square,  the 
height  of  one  being  4  feet  and  that  of  the  other  5  feet  ? 


234  SCHOOL  ARITHMETIC. 

37.  What  part  of  the  annual  revolution  does  the  earth 
make  in  24  da.  6  hr.  47  min.  ? 

38.  If  1  bu.  3  pk.  of  seed  is  required  to  plant  an  acre, 
how  much  must  be  sowed  upon  2  A.  16  sq.  rd.? 

39.  3960  cu.  yd.  of  earth  were  removed  in  digging  a  ditch 
360  rd.  long  and  6  ft.  deep.     How  wide  is  the  ditch  ? 

40.  A  druggist  pays  53|^  a  pound  avoirdupois  for  9  lb. 
of  borax.  If  he  sells  it  at  the  rate  of  6^^  an  ounce  apothe- 
caries', how  much  does  he  gain  ? 

41.  How  many  silver  forks,  each  weighing  2.3  oz.,  can  be 
made  from  9  lb.  18  pwt.  of  silver  ? 

42.  How  many  sq.  yd.  in  the  walls  and  ceiling  of  a  room 
24  ft.  long,  18  ft.  wide,  and  12  ft  high  ? 

43.  How  many  cords  of  wood  in  a  pile  336  ft.  by  9  ft.  by 
6ft.? 

44.  If  it  takes  a  man  |  of  a  day  to  mow  an  acre  of  grass, 
how  long  will  it  take  him  to  mow  3  A.  45  sq.  rd.  16  sq.  yd.? 

45.  If  ^1  of  a  bushel  of  salt  can  be  made  from  54  gal.  of 
salt  water,  how  much  salt  can  be  made  from  72  gal.? 

46.  A  plank  37  ft.  4  in.  long  and  4  in.  thick  contains  4f 
cu.  ft.     What  is  the  width  ? 

47.  A  man  who  had  2  miles  to  travel,  walked  5  rd.  7  ft. 
How  far  had  he  yet  to  travel  ? 

48.  If  5.3  T.  of  porcelain  clay  cost  $106,  what  is  the  cost 
of  438  pounds  ? 

49.  How  many  cubic  yards  of  air  in  a  room  18  ft.  by  15  ft. 
by  9  ft.  ? 

50.  What  must  be  the  depth  of  a  measure  18^  in.  square  to. 
contain  a  bushel  ? 

51.  If  4  bu.  3  pk.  4  qt.  1.6  pt.  of  wheat  makes  one  barrel 
of  flour,  and  the  toll  is  4  qt.  a  bushel,  how  many  bushels 
of  wheat  must  I  take  to  the  mill  in  order  to  get  five  barrels 
ot  flour  ? 

52.  When  corn  meal  is  selling  at  80^  per  cwt.,  how  many 
pounds  will  10^  buy  ? 


REVIEW   WORK.  235 

53.  A  building  lot  100  ft.  front  contains  1  acre.  How  deep 
is  the  lot  ? 

54.  How  many  cakes  of  maple  sugar  8  inches  by  6  inches 
by  3  inches  can  be  packed  in  a  box  24  inches  by  18  inches 
square  in  the  clear  ? 

55.  If  a  load  of  wood  is  8  ft.  long  and  3  ft.  wide,  how  high 
must  it  be  to  contain  a  cord  ? 

56.  When  each  end  and  the  middle  of  boards  IGJ  feet  long 
are  nailed  to  posts,  how  many  posts  are  in  a  fence  around  a 
field  30  rods  long  and  20  rods  wide  ? 

SUPPLEMENTARY   EXERCISES  (FOR   ADVANCED   CLASSES). 

346.  1.  How  many  secojids  in  the  month  of  February, 
1899  ? 

2.  How  long  will  it  take  a  body  moving  at  the  rate  of  a  mile 
a  minute  to  travel  from  the  earth  to  the  moon,  the  dis- 
tance being  239,000  miles  ? 

3.  How  many  seconds  in  the  circumference  of  a  silver  dollar? 

4.  Which  is  heavier,  and  how  much — a  pound  and  an 
ounce  of  gold,  or  a  pound  and  an  ounce  of  lead  ? 

5.  How  many  pounds  of  gold  are  equal  in  weight  to  G  lb. 
of  feathers  ? 

6.  How  many  bushels  of  corn  will  a  receptacle  contain  that 
holds  5,000  gallons  of  water  ? 

7.  How  long  will  it  take  to  count  a  million,  counting  80  a 
minute,  and  12  hours  a  day  ? 

8.  A  printer  used  3  reams,  5  quires,  19  sheets  of  paper  for 
printing  half-sheet  sale  bills.  How  many  did  he  print, 
allowing  1  quire  to  a  ream  for  waste  ? 

9.  There  are  9  oz.  of  iron  in  the  blood  of  one  man.  The 
blood  of  how  many  men  would  be  required  to  furnish  suffi- 
cient iron  to  make  a  kettle  weighing  22^  lb.? 

10.  A  merchant  bought  a  barrel  (42  gal.)  of  vinegar  for 
16.30.  How  much  water  must  be  added  to  reduce  the  first 
cost  to  10^  a  gallon  ? 


236  SCHOOL  ARITHiMETIC. 

11.  In  18}  carat  gold,  what  part  is  alloy  ? 

12.  How  many  silver  dollars  can  be  made  from  a  bar  of 
silver  weighing  11  lb.  9  oz.  avoirdupois  ? 

13.  How  many  tucks  ^  inch  wide  can  be  made  in  a  strip  of 
muslin  a  yard  long,  leaving  ^  of  an  inch  between  the  edge 
of  one  tuck  and  the  stitching  of  the  next  ? 


LONGITUDE    AND    TIME. 

347.  1.  Does  the  earth  rotate  from  west  to  east,  or  from 
east  to  west  ?     In  how  many  hours  does  it  rotate  once? 

2.  In  what  time  does  any  place  on  the  earth's  surface  pass 
through  360°  ?  Through  how  many  degrees  does  it  pass  in 
1  hour  ?    In  1  minute  ?    -^^  of  15°  =  (     )  ?    \°  =  how  many  '? 

3.  Since  the  earth  moves  15'  in  1  minute  of  time,  how  far 
does  it  move  in  1  second  of  time  ?     ^'  =  how  many  ."? 

4.  How  long  does  it  take  the  earth  to  turn  through  15°  ? 
30°?     60°?.    90°?     180°? 

5.  How  does  the  number  of  hours  compare  with  the 
number  of  degrees  traveled  ? 

6.  Which  city  has  sunrise  first — Baltimore  or  Chicago  ? 
Why?-  ^       ^ 

7.  If  my  watch  shows  the  correct  time  when  I  leave 
Boston,  will  it  be  too  slow  or  too  fast  when  I  reach  Denver  ? 

8.  A  man  from  Cleveland  arrived  at  another  city  and  found 
his  watch  20  minutes  too  slow.  Had  he  traveled  east  or 
west  ?     How  many  degrees  ? 

348.  A  Meridian  (mid-day  line)  is  an  imaginary  line 
running  north  and  south  from  pole  to  pole.  All  places  on 
the  same  meridian  have  their  mid-day,  or  noon,  at  the  same 
moment ;  that  is,  when  the  sun's  rays  are  vertical  on  that 
meridian. 

349.  The  Longitude  of  a  place  is  its  distance  (in  degrees, 
etc.)  east  or  west  from  a  given  meridian. 


LONGITUDE  AND  TIME.  ^37 

350.  The  given  meridian  from  which  longitude  is  gen- 
erally reckoned  is  called  the  Prime  Meridian.  It  passes 
through  Greenwich,  a  part  of  London. 

Notes. — 1.  From  the  prime  meridian,  longitude  is  reckoned  east  and 
west  to  180°.  West  longitude  is  designated  by  the  letter  W;  east  lon- 
gitude by  the  letter  E. 

2.  Clocks  show  later  (faster)  time  at  places  east  of  a  given  place,  and 
earlier  (slower)  time  at  places  west  of  a  given  place. 

351.  Since  the  earth  turns  upon  its  axis  once  in  24  hours, 
any  place  on  the  earth's  surface  passes  through  360°  in  that 
time.     Hence  we  deduce  the  following 

Table  of  Relations. 

LONGITUDE.  TIME. 

360°  corresponds  to  24  hours. 
15°  ''  ''    1  hour. 

15'  ''  «    1  minute. 

15"  "  "    1  second. 

1°  "  "    -^^  h\\,  or  4  min. 

1'  ''  "    i^jr  min.,  or  4  sec. 

352.  To  find  the  difference  In  time  between  two 
places,  the  difference  in  longitude  being  given. 

WRITTEN     EXERCISES. 

1.  Boston  is  71°  3'  30"  west  of  London.  What  is  the  dif- 
ference in  time  ? 

15)71°      3'    30"  It  will  be  seen  by  the  table  that  the  numbers  de- 

7     77     ^i        noting  the  difference  in  time  between  two  places 

are  (V  of  those  denoting  longitude.     The  process  is 

the  same  as  in  division  of  compound  numbers,  and  the  difference  in  time 

is  4  hours,  44  minutes,  and  14  seconds. 

2.  The  difference  in  longitude  between  two  places  is  50°  25'. 
What  is  the  difference  in  time  ? 

3.  Cincinnati  is  84°  29'  45"  west  from  Greenwich.  Find 
the  difference  in  time. 


238  SOflOOL  ARtTHMETia 

4.  When  it  is  noon   at   Cincinnati,  what  is  the  time  at 
Greenwich  ? 

SHORT  LIST  OF    CITIES,    WITH    LONGITUDE    FROM    GREENWICH. 
CITIES.  LONGITUDE.  CITIES.  LONGITUDE. 

Greenwich 0°    0'    0"  St.  Louis 90°  15'  15 "      W. 

New  York 73°  58'  25.5'  W.  San  Francisco  .  .122°  25'  40.8  "  W. 

Paris    2°  20'  15"  E.  St.  Petersburg ..  30°  19'    0"       E. 

Boston 71°    3'  30"  W.  Richmond 77°  26'    4"      W. 

Chicago 87°  36'  42"  W.  Denver 104°  59'  33"      W. 

Rome 12°  27'  14"  E.  Pittsburg 80°     2'     0"      W. 

Cleveland 81°  40'  30"  W.  Honolulu 157°  51'  48"      W. 

Washington ....  77°     3'     0"  W.  Jerusalem 35°  13'  25"       E. 

Calcutta 88^  20'     8"  E.  Athens. 23°  43'  55.5"    E. 

Find  the  difference  in  time  between  : 
6.  Boston  and  Pittsburg. 

6.  Chicago  and  AVashington. 

7.  Eome  and  Calcutta. 

8.  San  Francisco  and  Denver. 

9.  Philadelphia  and  Cleveland. 

10.  New  York  and  Paris. 

11.  Richmond,  and  St.  Petersburg. 

12.  Honolulu  and  Jerusalem. 

13.  When  it  is  noon  at  New  York,  what  is  the  time  at 
Denver  ? 

14.  Would  a  traveler's  watch  be  too  fast  or  too  slow,  and 
how  much,  when  he  goes  from  St.  Louis  to  Athens  ? 

353.  To  find  the  difference  in  longitude  between 
two  places,  the  difference  in  time  being  given. 

1.  When  it  is  noon  at  San  Francisco,  it  is  8  minutes  45 
seconds  past  2  p.m.  at  St.  Louis.  What  is  their  difference 
in  longitude  ? 

It  will  be  seen  by  the  table  that  there  are  15  times  as 
many  °,  ',  and  "  of  longitude  between  two  places  as  there 
are  hours,  minutes,  and  seconds  of  time.    The  process  is 
32     11     15       ^^®  same  as  in  multiplication  of  compound  numbers, 
and  the  difference  in  longitude  is  33°  11'  15". 


hr. 

rain.  sec. 

2 

8    45 

15 

LONGITUDE  AND  TIME.  239 

a.  What  is  the  difference  in  longitude  between  two  places 
whose  difference  in  time  is  4  hours,    12  minutes,  and   23 

seconds  ? 

Find  the  difference  in  longitude  between  : 

3.  New  York  and  Chicago. 

4.  Philadelphia  and  Cleveland. 
6.  Richmond  and  Denver. 

6.  Cleveland  and  Greenwich. 

7.  Boston  and  San  Francisco. 

8.  Washington  and  Paris. 

9.  Pittsburg  and  Calcutta. 

10.  Rome  and  St.  Petersburg. 

11.  When  a  traveler  reached  Boston  he  found  that  his 
watch  was  4  hr.  25  min.  too  fast.  What  was  the  longitude 
of  the  place  from  which  he  started  ? 

12.  When  it  is  12  o'clock  M.  at  Pittsburg,  it  is  9  hr.  10 
min.  18  sec.  a.m.  at  Portland,  Oregon.  What  is  the  longi- 
tude of  Portland  ? 

13.  Mr.  R  left  Philadelphia  and  traveled  eastward  at  the 
rate  of  ^°  an  hour.  How  much  too  slow  was  his  watch  at 
the  end  of  two  weeks  ? 

14.  The  difference  in  time  between  two  places  on  the 
equator  is  2  hr.  45  min.  30  sec.  In  how  many  hours  could  a 
railroad  train  run  from  one  place  to  the  other  at  the  rate  of 
30  geographic  miles  an  hour  ? 

15.  Midnight  comes  1  hr.  5  min.  42  sec.  sooner  at  Quebec 
than  at  Chicago.     What  is  the  longitude  of  Quebec  ? 

16.  At  Richmond  the  sun  rises  1  hr.  2  min.  52  sec.  earlier 
than  at  St.  Paul,  and  2  hr.  59  min.  49  sec.  earlier  than  at 
San  Francisco.  What  is  the  difference  in  longitude  between 
St.  Paul  and  San  Francisco  ? 

354.  To  find  the  difference  in  the  longitude  of  two  places  : 

1.  If  both  longitudes  are  east,  or  if  both  are  west,  sub- 
tract ;  if  one  is  east  and  the  other  ivest,  add. 


240  SCHOOL  ARITHMETIC. 

2.  If  the  sum  of  two  longitudes  is  greater  than  180°,  the 
sum  must  he  subtracted  from  360°  to  obtain  the  correct  differ- 
ence of  longitude. 

STANDARD    TIME. 

355.  The  time  we  have  been  considering  is  called  Local 
Time,  which  is  determined  by  the  rotation  of  the  earth  on 
its  axis.  To  avoid  the  confusion  and  mistakes  incident  to 
such' time,  the  railroad  companies  have  adopted  the  time  of 
the  meridians  of  75°,  90°,  105°,  and  120°  as  the  standards 
by  which  to  run  their  trains.  This  railroad  time  is  called 
Standard  Time. 

356.  This  new  plan  divides  the  United  States  and  Canada 
into  4  belts,  extending  north  and  south,  each  about  15°  wide. 
All  places  in  the  same  belt  have  the  same  time,  regardless  of 
their  longitude. 

Note. — The  division  lines  of  the  time  belts  are  not  exactly  7^°  on  either 
side  of  the  hour  meridians,  but  are  somewhat  irregular,  passing  through 
leading  railway  termini. 

^^  Have  the  pupil  turn  to  map  in  his  geography  showing  the  stand- 
ard time  belts.     Which  is  the  widest  ? 

357.  The  first  belt  lies  on  both  sides  of  the  meridian  of 
75°,  which  passes  1°  west  of  ^"ew  York  city,  and  all  places 
therein  have  the  time  of  that  meridian,  which  is  called  East- 
ern time. 

Thus,  when  a  man  goes  from  Richmond  to  New  York,  Boston,  or 
Quebec,  he  finds  that  his  watch  is  neither  too  fast  nor  too  slow. 

358.  Tlie  second  belt  lies  on  both  sides  of  the  meridian 
of  90°,  which  passes  near  New  Orleans  and  St.  Louis.  The 
time  in  this  belt  is  called  Central  time,  and  is  1  hour  slower 
than  Eastern  time. 

Query. — Why  is  Eastern  time  just  1  hour  faster  than  Central  time  ? 

359.  The  third  belt  lies  on  both  sides  of  the  meridian  of 


LONGITUDE  AND  TIME.  241 

105°,  wliich  passes  near  Pike's  Peak.  The  time  in  this  belt 
is  called  Mountain  time,  and  is  1  hour  slower  than  Central 
time,  and  2  hours  slower  than  Eastern  time. 

Query.— A  man  who  goes  from  Baltimore  to  Denver  will  find  his 
watch  how  much  too  fast  ? 

360.  The  fourth  belt  lies  on  both  sides  of  the  meridian  of 
120°,  which  passes  9  degrees  east  of  San  Francisco.  The 
time  there  is  3  hours  slower  than  Eastern  time,  and  1  hour 
slower  than  Mountain  time.     It  is  called  Pacific  time. 

Query. — When  it  is  9  a.m.  at  San  Francisco,  what  is  the  time  at  New 
York  ?    At  Leadville  ?    At  St.  Paul  ? 

COMPARISON   OF  TIMES. 
BOSTON  CHICAGO  DENVER         PORTLAND  (OR.) 

Standard  Time,  ]2  m.      11  a.m.  10  a.m.  9  a.m. 

Local  Time,         12  m.     10:53}  a.m.     9:44^  a.m.     8:34|a.m. 

^^  On  the  4  standard  meridians,  local  time  and  standard  time  are 
the  same.  In  other  places  standard  time,  being  fixed  arbitrarily,  is  not 
the  correct  or  local  time. 

WRITTEN    EXERCISES. 

361.  1.  At  Pittsburg,  what  is  the  difference  between 
local  time  and  standard  time  ? 

Longitude  of  Pittsburg,  80°  2'  0"  W. 

Longitude  of  the  meridian  of  75°,    75°  0'  0"  W. 

15)  5°  2'  0" 


20  8 

Pittsburg  has  the  time  of  the  meridian  of  75"  instead  of  that  of  her 
own  meridian.  The  difference  in  longitude  between  the  two  meridians 
is  readily  found,  and  the  difference  in  time  is  ^  as  great.  Hence,  stand- 
ard time  at  Pittsburg  is  20  min.  8  sec.  faster  than  local  time. 

2.  What  is  the  difference  at  New  York  between  standard 
and  local  time  ? 

3.  When  it  is  3  p.m.,  standard  time,  at  San  Francisco, 
what  is  the  local  time  there  ? 

10 


242  SCHOOL  ARITHMETIC. 

4.  The  longitude  of  Staunton,  Va.,  is  79°  4'  15".  When 
it  is  5  A.M.  there,  standard  time,  what  is  the  local  time  ? 

6.  When  it  is  12  m.  at  Paris,  Avhat  is  the  standard  time  at 
New  York  ?     The  local  time  ? 

6.  What  is  the  local  time  at  Denver  when  the  standard 
time  at  Cleveland  is  9  a.m.  ? 

I>ATE    LINE. 

362.  The  boundary  line  between  adjoining  regions  in 
which  the  calendar  day  is  different  is  called  a  Date  Line. 
This  line  as  now  agreed  upon  nearly  coincides  with  the  me- 
ridian of  180°. 

Note. — Suppose  a  man  to  leave  London  at  noon  on  Friday  and  travel 
westward  just  as  fast  as  the  earth  rotates.  He  keeps  the  sun  directly 
overhead  all  the  time,  and  it  seems  to  him  that  it  is  still  Friday  noon 
when  he  reaches  London  again  24  hours  after  starting,  when  in  reality 
it  is  Saturday  noon.  He  has  lost  a  day  in  his  reckoning.  Had  he  trav- 
eled eastward  he  would  have  gained  a  day.  There  is  the  same  loss  and 
gain  wheii  traveling  is  done  at  ordinary  speed. 

Hence  is  seen  the  necessity  for  a  fixed  place  at  which  each  new  day 
shall  begin.  This  place  is  the  date  line  ;  there  a  new  day  begins  every 
midnight.  Thus,  July  4th  begins  on  the  meridian  of  180°  at  midnight 
following  the  3d  of  July.  At  that  time  it  is  midday,  July  3,  at  London 
(Greenwich)  ;  12  hours  later  it  is  midnight  at  London,  and  that  city 
enters  upon  the  new  date — July  4  ;  it  is  then  noon  of  the  4th  on  the  date 
line,  while  at  Chicago  it  is  about  6  p.m.,  July  3. 

Each  day  has  the  same  date  all  around  the  earth.  Thus,  Sunday, 
May  1,  1899,  began  at  midnight  on  the  date  line,  and  thereafter  as  each 
place  on  the  earth  had  midnight  it  began  to  record  the  date  as  "Sun- 
day, May  1." 

It  is  a  day  later  on  one  side  of  the  date  line  than  it  is  on  the  other. 
Thus,  when  it  is  Monday  in  San  Francisco  it  is  Tuesday  in  Hong  Kong. 
Hence  navigators  change  their  calendar  one  day  in  crossing  this  date  line. 
Going  east,  they  count  the  same  day  twice  ;  going  west,  they  skip  a  day. 

Queries. — 1.  When  it  is  noon,  Jan.  1,  1900,  at  New  York, 
what  is  the  date  and  time  on  the  island  of  Luzon,  longitude 
120°  E.  ? 


LONGITUDE  AND  TIME.  243 

2.  When  it  is  Saturday  noon  at  Athens,  what  is  the  day 
and  local  time  at  Chicago  ? 

3.  When  it  is  2  a.m.  on  Friday  at  Calcutta,  what  is  the 
day  and  hour  at  Boston  ? 

SUPPLEMZNTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

363.  1.  If  A  leaves  home  at  12  m.  on  Monday,  and  on 
Saturday  finds  his  watch  1  hr.  15  min.  slow,  in  what  direc- 
tion and  how  far  has  he  traveled  ? 

2.  A  mail  starts  from  Philadelphia  and  travels  westward. 
When  he  stops,  he  ascertains  that  his  watch  is  6  hr.  slow.  In 
what  longitude  did  he  stop  ?  Through  how  many  degrees 
did  he  travel  ? 

3.  A  man  traveling  eastward  from  a  point  on  the  equator 
stops  when  his  watch  has  become  2  hr.  40  min.  slow.  How 
many  miles  has  he  traveled  ? 

4.  If  a  man  starts  westward  from  Paris  and  travels  181°, 
will  his  watch  be  too  fast  or  too  slow,  and  how  much  ? 

5.  A  and  B  start  from  different  points  and  travel  towards 
each  other.  When  they  meet,  A's  watch  is  40  minutes  slow, 
and  B's  is  1  hour  fast.  How  far  apart  are  the  starting-points  ? 
In  what  direction  did  each  travel  ? 

6.  When  it  is  noon,  local  time,  at  New  York,  on  what 
meridian  is  it  midnight  ? 

7.  When  it  is  Sunday  noon,  local  time,  at  Chicago,  what  is 
the  day  and  time  at  Jerusalem  ? 

8.  A  vessel  had  sailed  a  certain  distance  on  a  parallel  of 
latitude  when  the  captain  found  that,  although  the  sun  was 
on  the  meridian,  his  Greenwich  chronometer  indicated  1:47 
P.M.     What  was  the  ship's  longitude  ? 

9.  The  battle  of  Manila  began  at  6  a.m..  May  1.  Had 
Dewey  cabled  at  once  to  Washington,  at  what  hour  and  on 
what  date  would  the  message  have  reached  the  President, 
allowing  1  hour  for  transmission,  and  considering  the  longi- 
tude of  Manila  to  be  120°  E.  ? 


PRACTICAL    MEASUREMENTS. 
painti:ng  and  plastering. 

364.  Painting  and  plastering  are  usually  estimated  by  the 
square  yard. 

Allowances  for  openings  ar&  sometimes  made,  but  there  is 
no  fixed  rule  as  to  how  much  should  be  deducted. 

1.  Find  the  cost  of  painting  the  outside  of  a  house  36  ft. 
long,  28  ft.  wide,  and  24  ft.  high,  at  $.25  a  sq.  yd. 

2.  A  room  18  ft.  long,  15  ft.  wide,  and  12  ft.  high  has  2 
doors,  each  8  ft.  by  4  ft.  6  in.  Find  the  cost  of  plastering 
the  walls  and  ceiling  at  $.35  a  sq.  yd.,  making  allowance  for 
the  doors. 

3.  A  parlor  is  15  ft.  9  in.  square,  and  a  reception-room  is 
12  ft.  by  16  ft.  The  height  of  each  is  11  ft.  6  in.  What  will 
1)6  the  cost  of  plastering  the  walls  and  ceilings  at  $.33  a 
sq.  yd.  ? 

4.  How  much  would  be  saved  by  having  the  ceilings  kal- 
somined  at  a  cost  of  $.18  a  sq.  yd.  ? 

5.  Measure  your  schoolhouse,  and  prepare  a  problem  in 
painting  for  your  class. 

6.  Find  the  cost  of  wainscoting  a  hall  30  ft.  long,  8  ft. 
wide,  and  12  ft.  8  in.  high,  at  $.40  a  sq.  yd. 

7.  Measure  your  schoolroom,  and  prepare  a  problem  in 
plastering  for  your  class. 

8.  Find  the  cost  of  painting  a  gable  roof  45  ft.  long  and 
24  ft.  wide,  at  28  ct.  a  sq.  yd. 


MtiAsiJRtiM^NT    <^P   LiltMBElt. 

365.  When  boards  are  1  inch  thick  or  less,  they  are  esti- 
mated by  the  square  foot  of  surface,  the  thickness  not  being 
considered. 

Thus,  a  board  8  ft.  long,  1  ft.  wide,  and  1  inch  (or  less)  thick  con- 
tains 8  square  feet.     Its  surface  is  a  rectangle. 

366.  A  board  1  foot  square  and  1  inch  tliick  is  called  a 
board  foot,  or  a  foot  board  measure. 

When  lumber  is  more  than  1  inch  thick,  the  number  of 
board  feet  depends  upon  the  thickness. 

Thus,  a.  board  8  feet  long,  1  foot  wide,  and  2  inches  thick  contains 
16  board  feet,  or  twice  as  many  as  if  only  1  inch  thick. 

1.  How  many  square  feet  in  a  board  12  ft.  long  and  12  in. 
wide  ?     How  many  board  feet  in  it  if  it  is  1  inch  thick  ? 

2.  How  many  feet  board  measure  in  a  board  16  ft.  long 
and  15  in.  wide  if  it  is  1  inch  thick  ?  How  many  if  it  is  ^ 
inch  thick  ? 

3.  Find  the  cost  of  a  board  14  ft.  long,  6  in.  wide,  and  an 
inch  thick,  at  3^  a  board  foot. 

4.  An  inch  board  18  ft.  long  is  16  in.  wide  at  one  end  and 
8  in.  wide  at  the  other.    How  many  board  feet  does  it  contain  ? 

The  average  width  of  a  board  that  tapers  uniformly  is  one-half  the 
sum  of  the  end  widths.     A  board  like  this  has  the  form  of  a  trapezoid. 

5.  A  half-inch  board  16  ft.  long  is  16  in.  wide  at  one  end 
and  tapers  to  a  point  at  the  other.  What  is  it  worth  at  3^^ 
a  foot  board  measure  ? 

What  is  the  form  of  this  board  ? 

6.  How  many  feet  board  measure  in  a  beam  24  ft.  long,  15 
in.  wide,  and  3  in.  thick  ? 

24  X  IJ  =  30,  the  number  of  sq.  ft.  in  the  surface,  or  the  number  of 
board  feet  in  1  inch  of  the  thickness.  Hence,  in  3  inches  of  thickness 
there  are  3  x  30  board  feet,  or  90  board  feet. 

Jn  any  board  or  timber  the  number  of  board  feet  in  1  inch 


246  SCHOOL  ARITHMETIC. 

of  thichness  is  equal  to  the  number  of  square  feet  in  the  sur- 
face of  one  side. 

Hence,  to  find  the  number  of  feet  boiird  measure,  when 
lumber  is  more  than  1  inch  thick, 

EuLE. — Multiply  the  number  of  square  feet  in  the  surface 
of  one  side  by  the  number  representing  the  thickness  in  inches. 

7.  How  many  feet  board  measure  in  a  plank  14  ft.  long, 
16  in.  wide,  and  2  in.  thick  ? 

8.  How  many  board  feet  in  a  log  18  ft.  long,  15  in.  wide, 
and  12  in.  thick  ? 

9.  Mr.  H  bought  30  joists,  each  20  ft.  long,  8  in.  wide,  and 
3  in.  thick,  at  118.50  per  M.     Find  the  cost. 

•  It^f^In  lumber  measure  "  per  M  "  means  "by  the  thousaml"  (board 
feet). 

10.  What  will  it  cost  to  floor  a  two-story  warehouse,  24  ft. 
by  36  ft.,  with  two-inch  planks,  at  $25  per  M  ? 

11.  How  many  feet  board  measure  in  a  piece  of  timber  32 
ft.  long,  18  in.  wide  at  one  end  and  6  in.  wide  at  the  other, 
the  thickness  being  14  inches  ? 

12.  Two  men  bought  a  piece  of  timber  10  ft.  long  and  15 
in.  square,  paying  at  the  rate  of  $24  per  M.  How  much 
should  each  man  pay  ? 

13.  A  farmer  used  inch  boards  to  make  a  feed-box  which 
measured  on  the  outside  8  ft.  in  length,  3  ft.  2  in.  in  width, 
and  4  ft.  9  in.  in  height.  What  was  the  cost  of  the  boards, 
at  $18  per  M  ? 

14.  James  McKnight  bought  from  Jas.  Laird,  Charleston, 
S.  C,  as  follows  : 


40  joists,         2x6, 

18  ft. 

long,  @  $25, 

16  beams,       6x9, 

20  " 

"      ''     30, 

72  scantling,  2x4, 

12  '' 

''      "     24, 

240  boards,    1  x  10, 

12  '' 

''      ''     18, 

24  planks,      2  x  14, 

16  '' 

''      ''     17.50. 

Make  out  a  complete  bill,  and  find  the  amount  due  Laird. 


PRACTICAL  MEASUREMENTS.  247 

BRICK  WORK  AND  STONE  WORK. 

367.  Brick  work  is  commonly  estimated  by  the  thousand 

bricks  ;  stone  work  by  i\\Q  perch ,  which  contains  24J  cu.  ft. 

The  number  of  bricks  in  a  cubic  foot  of  wall  depends  upon 
the  size  of  the  bricks.  Common  bricks  are  8  in.  x  4  in.  x  2 
in.,  and  22  of  these  are  assumed  to  build  1  cubic  foot  of 
wall. 

■  In  measuring  walls  of  buildings,  masons  and  bricklayers 
take  the  entire  outside  length,  thus  measuring  the  corners 
twice.     This  must  not  be  done  in  estimating  material. 

In  estimating  the  tvorky  allowance  for  openings  in  the 
walls  is  sometimes  made,  but  the  amount  to  be  deducted,  if 
any,  should  be  specified  in  a  written  contract. 

In  estimating  the  material,  all  openings  should  be  deducted. 
In  stone  work  \  is  allowed  for  mortar  and  filling.  In  a 
perch  of  masonry,  without  openings,  there  are  only  22  cu.  ft. 
of  stone. 

1.  If  22  common  bricks  build  a  cubic  foot  of  wall,  how 
many  bricks  will  be  required  to  build  a  wall  containing  1000 
cu.  ft.  ? 

2.  How  many  common  bricks  in  a  wall  36  ft.  long,  7  ft. 
high,  and  16  in.  thick  ? 

3.  How  many  perches  of  masonry  in  the  walls  of  a  cellar 
that  is  40  ft.  long  and  24  ft.  wide,  the  walls  being  9  ft.  high 
and  18  in.  thick  ? 

4.  How  many  perches  of  stone  in  the  walls  of  the  above 
cellar  ? 

5.  Find  the  cost  of  building  said  walls  at  $1.50  a  perch. 
€.  Had  the  walls  been  built  at  10^  a  cu.  ft.,  how  much 

more  or  less  would  the  mason  have  received  ? 

7.  The  inside  length  of  a  storeroom  is  56  ft.,  the  width  44 
ft.  The  outside  length  is  60  ft.,  the  width  48  ft.  If  the 
walls  are  22  ft.  high,  how  many  cubic  feet  do  they  contain, 
allowing  nothing  for  openings,  and  counting  corners  once  ? 


248  SCHOOL  ARITHMETIC. 

8.  Mr.  S  built  a  house  38  ft.  long,  32  ft.  wide,  and  26  ft. 
high.  The  front  wall  is  built  of  stone,  and  is  2  ft.  thick. 
The  others  are  built  of  common  bricks,  and  are  16  in.  thick. 
There  are  16  windows,  each  6  by  3  ft.,  and  4  doors,  each  7 
by  3^  ft.  In  the  front  wall  are  4  windows  and  one  door. 
How  many  bricks  and  perches  of  stone  were  used  ? 

CARPETING. 

368.  The  amount  of  carpet  that  must  be  bought  for  a 
room  depends  upon  the  length  and  number  of  strips,  and 
the  waste  in  matching  the  patterns. 

The  number  of  strips  often  depends  upon  whether  they  are  laid 
lengthwise  or  crosswise.  Thus,  in  a  room  15  by  17,  5  strips,  each  a  yard 
wide,  will  be  sufficient,  if  laid  lengthwise.  If  laid  the  other  way,  6 
strips  will  be  required,  and  one  foot  of  width  may  be  turned  under  or 
cut  off. 

5  X  17  ft.  =  85  ft.,  or  28i  yd.  when  laid  lengthwise. 

6  X  15  ft.  =  90  ft.,  or  30  yd.         "       "     crosswise. 

The  waste  in  matching  tlie  patterns  cannot  be  estimated  except  by 
actual  measurement  of  the  carpet.  In  the  following  problems  waste  in 
matching  is  not  considered. 

1.  A  schoolroom  is  30  ft.  long  and  26  ft.  wide.  How  much 
matting  a  yard  wide  should  be  bought  to  cover  the  floor  if 
the  strips  are  to  be  laid  lengthwise  ? 

Suggestions. — 1.  How  wide  is  the  room  ?  Then  how  many  strips  will 
be  needed  ?  2.  Can  we  buy  §  of  a  strip  (in  width)  ?  Then  how  many 
strips  must  be  bought  ?  What  may  we  do  with  the  surplus  width  ?  3. 
How  long  must  each  strip  be  ?    Then  how  many  yards  must  be  bought  ? 

2.  A  dining-room  is  18  ft.  6  in.  long  and  15  ft.  wide.  How 
much  ingrain  carpet  a  yard  wide  must  be  bought  to  cover  it 
if  the  strips  are  to  be  laid  lengthwise  ? 

3.  What  will  be  the  cost  of  carpeting  a  room  14  ft.  by  18  ft. 
with  Brussels  carpet  27  in.  wide,  laid  lengthwise,  if  3  yards 
cost  $4.50  ? 

4.  A  barber  shop  15  ft.  by  17  ft.  8  in.  is  covered  with  oil- 


PRACTICAL  MEASUREMENTS.  249 

cloth  2  yd.  wide,  laid  across  the  room.     What  was  the  cost 
of  the  oilcloth  if  2^  yd-,  cost  $2.25  ? 

5.  In  a  parlor  14  ft.  by  20  ft.  the  carpet  is  J  yd.  wide,  and 
is  laid  crosswise.     Find  its  cost  at  $.85  a  yard. 

6.  My  library,  which  is  12  ft.  square,  is  covered  with  Cliina 
matting  36  in.  wide,  for  which  I  paid  $.05  a  yard.  IIow 
much  did  the  matting  cost  me  ? 

7.  Find  the  cost  of  a  rug  for  a  room  3  yd.  2  ft.  8  in.  by  4  yd. 
@  $1.25  a  sq.  yd. 

8.  How  many  tiles  8  in.  square  will  be  required  to  lay  a 
floor  12  ft.  by  32  ft.  ? 

9.  Brussels  carpet  27  in.  wide  is  laid  lengthwise  on  the  floor 
of  Mr.  B's  reception-room,  which  is  13  ft.  wide.  One  yard 
cost  $1.75,  and  the  entire  cost  was  $52.50.  What  was  the 
length  of  the  room  ? 

PAPERING. 

369.  The  amount  of  wall  paper  required  to  paper  a  room 
depends  upon  the  area  of  the  walls  and  ceiling  and  the  waste 
in  matching. 

(a).  Wall  paper  is  put  up  in  double  rolls,  but  the  prices 
quoted  are  for  single  rolls. 

(b).  A  single  roll  contains  8  yards,  18  inches  wide,  its  area 
being  36  square  feet.  Allowing  for  all  waste,  this  will  cover 
30  square  feet  of  wall. 

(c).  In  estimating  the  number  of  rolls  required  for  a  room, 
some  dealers  deduct  the  exact  area  of  the  doors  and  windows, 
while  others  deduct  an  approximate  area,  allowing  20  square 
feet  for  each. 

(d).   Borders  vary  in  width,  and  are  sold  by  the  yard. 

(e).  The  number  of  single  rolls  required  for  the  walls  and 
that  for  the  ceiling  must  be  estimated  separately,  and  can  be 
found  only  approximately.  Two  methods  employed  by  dealers 
are  shown  in  (a)  and  (J)  below. 


250  SCHOOL  ARITHMETIC. 

.  Example. — Room  12  x  14, 10  feet  liigh,  one  window  6x4, 
one  door  7x4. 


(&) 
Area  of  walls 520  sq.  ft. 

Two  openings,  20  sq.  ft. 

each J^   "    " 

Difference : 480   "    " 

480  -r-  30  =:  16,  number  of  single 
rolls  needed.  This  method  allows 
for  waste  and  matching. 


(«) 
Area  of  walls 520  sq.  ft. 

Area  of  openings 52    "    " 

Difference 468    "    " 

468  -f-  36  =  13,  number  of  single 
rglls  required.  No  allowance  is 
here  made  for  matching  and  waste. 

The  area  of  the  ceiling  divided  by  36  (or  by  30  to  allow  for  waste) 
gives  the  number  of  rolls  required  for  that  ceiling.  The  number  of  yards 
of  border  required  is  the  same  as  the  perimeter  of  the  room  in  yards. 

1.  How  many  single  rolls  of  wall  paper  will  be  required  to 
paper  the  walls  and  ceiling  of  a  room  17  ft.  long  and  15  ft. 
wide,  not  allowing  for  waste,  the  height  from  baseboard  to 
ceiling  being  9  ft.  ? 

2.  Making  no  allowance  for  waste,  what  will  be  the  cost  of 
paper  and  border  for  the  walls  of  a  room  22  ft.,  long,  16  ft. 
6  in.  wide,  and  14  ft.  high,  if  paper  costs  $.75  a  roll  and 
border  $.  60  a  yard  ? 

3.  Measure  your  sitting-room  at  home,  and.  prepare  a 
problem  in  papering  for  your  class. 

4.  Estimate  the  cost  of  paper  for  your  parlor  at  $.75  a  roll. 

5.  My  dining-room  is  16  ft.  long,  13  ft.  9  in.  wide,  and 
10  ft.  high.  It  has  two  windows,  each  7  ft.  by  4  ft.,  and  a 
door  8  ft.  by  4J  ft.  Estimate  the  number  of  single  rolls 
required  to  paper  walls  and  ceiling,  allowing  for  waste. 

BINS,    CISTERNS,    ETC. 

370.  The  method  of  computing  the  contents  of  a  rect- 
angular solid  was  learned  in  Art.  327.  The  number  of  cubic 
units  in  a  box  or  cistern  is  found  in  the  same  manner. 

Note. — Cubic  inches  can  readily  be  reduced  to  bushels  or  gallons  if 
we  bear  in  mind  that 

2150.42  cu.  in.  =  1  bushel. 
231  cu.  in.-  =  1  gallon. 


PRACTICAL  MEASUREMENTS.  251 

1.  How  many  bushels  of  oats  will  be  required  to  fill  a 
feed-box  6  ft.  long,  3  ft.  wide,  and  4  ft.  deep  ? 

Find  the  contents  in  bushels  : 

2.  A  box  4  ft.  long,  3  ft.  wide,  and  2  ft.  deep. 

3.  A  bin  7  ft.  long,  4  ft.  wide,  and  5  ft.  high. 

4.  A  granary  18  ft.  long,  6  ft.  9  in.  wide,  and  5  ft.  G  in. 
high. 

Find  contents  in  gallons  : 

6.  A  cistern  5  ft.  long,  4  ft.  wide,  and  6  ft.  deep. 

6.  A  trough  8  ft.  long,  12  in.  wide,  and  8  in.  deep. 

7.  A  tank  6  yd.  long,  6  ft.  wide,  and  6  in.  deep. 

8.  A  bin  8  ft.  long,  4^  ft.  wide,  and  5  ft.  high  is  half  full 
of  wheat.     How  much  is  the  wheat  worth  at  $.80  a  bushel  ? 

9.  How  many  gallons  will  fill  a  tank  4  ft.  square  and  6  ft. 
deep  ? 

10.  My  cistern,  which  is  6  ft.  long,  5  ft.  wide,  and  8  ft. 
deep,  is  ^  full  of  water.  If  we  use  2  bbl.  of  the  water,  how 
many  gallons  will  remain  in  the  cistern  ? 

11.  Fifty  bushels  of  rye  are  in  a  bin  4  ft.  6  in.  square.  If 
the  bin  is  6  ft.  high,  how  much  more  rye  will  it  hold  ? 

12.  A  cistern  6  ft.  by  4  ft.  will  contain  1000  gallons.  How 
deep  is  it  ? 

13.  A  wagon-box  is  11|  ft.  long  and  ^  feet  wide.  When 
even  full  of  oats  it  contains  64.686  bushels.    What  is  its  depth  ? 

14.  A  shed  is  8  ft.  long,  6  ft.  wide,  and  6  ft.  4  in.  high. 
How  many  bushels  of  coal  will  it  hold  ?    How  many  of  oats  ? 

15.  A  box  3  ft.  long,  2  ft.  8  in.  wide,  and  2  ft.  high  is  j 
full  of  wheat.     What  is  the  weight  of  the  wheat  ? 

16.  A  tin  box  is  1  foot  square  and  1  foot  deep.  Find  the 
number  of  gallons  it  will  contain,  correct  to  3  decimal  places, 
and  reduce  the  decimal  to  qt.  and  pt. 

17.  A  tank  12  ft.  long  by  8  ft.  8  in.  wide  is  full  of  oil.  How 
many  gallons  must  be  drawn  off  to  lower  the  surface  3  ft.  ? 
How  many  barrels  ? 


THE  METKIC  SYSTEM. 

371.  The  uniform  system  of  measures  expressed  in  the 
decimal  scale,  which  was  first  adopted  in  France  in  1795,  is 
called  the  Metric  System.  It  is  used  in  nearly  all  countries 
of  continental  Europe  and  South  America,  especially  in 
scientific  work.  An  act  of  Congress  authorizes  its  use  in  the 
United  States. 

372.  The  system  is  based  on  the  principal  unit  of  length, 
called  the  Meter  (meaning  measure).  The  original  standard 
meter  is  a  rod  of  platinum  which  is  preserved  at  Paris  by  the 
French  government.  It  was  intended  to  be  .0000001  of  the 
distance  from  the  equator  to  the  pole,  but  more  careful  meas- 
urements show  this  distance  to  be  10,001,887  meters. 

373.  The  names  of  the  higher  and  lower  units  are  formed 
by  attaching  certain  prefixes  to  the  names  of  the  princip;il 
units.     These  prefixes  are,  in  part,  as  follows : 

(Greek)  (Latin) 

deka,    meaning  10  deci,  meaning  .1 

hekto,       "  100  centi,       ''         .01 

kilo,  ''        1000  milli,      "        .001 

myria,       ''        10000 


Table  of 

Length. 

A  myriameter 

= 

10,000 

meters. 

A  Mlometer  (Km.) 

= 

1000 

t( 

A  hektometer  (Hm. 

)  = 

100 

a 

A  dekameter  (Dm.) 

= 

10 

Si 

Meter  (m.) 

A  decimeter  (dm.) 

= 

.1     of  i 

a,  meter. 

A  centimeter  (cm.) 

= 

.01 

a 

A  millimeter  (mm.) 

— 

.001 

a 

THE  METRIC  SYSTEM. 


253 


This  table  may  be  arranged  thus  : 

10  millimeters   =  1  centimeter. 

10  centimeters  =  1  decimeter. 

10  decimeters    =  1  meter. 

10  meters  =  1  dekameter. 

10  dekameters   =  1  hektometer. 

10  hektometers  =  1  kilometer. 

10  kilometers     =  1  myriameter. 
Notes. — 1.  As  in  U.  S.  money  we  seldom  speak  of  anything  except 
dollars  and  cents,  so  in   the  metric  system  only  the  units  printed  in 
italics  are  commonly  used. 

2.  In  practice,  length  values  are  read  in  three  denominations.  Thus, 
1  dm.  5  cm.  is  read  fifteen  centimeters.  Values  inconveniently  large  to 
be  expressed  in  meters  are  read  as  kilometers. 

3.  A  length  given  in  one  unit  may  be  changed  to  another  by  simply 
moving  the  decimal  point  the  requisite  number  of  places.  Thus,  75  dm. 
=  7.5  m.,  and  75  cm.  =  .75  m. 

374.  The  annexed  scale  shows  the  decimeter  divided  into 
centimeters,  and  the  latter  into  millimeters.  It  also  com- 
pares the  decimeter  with  four  inches.  The  teacher  should 
by  all  means  have  a  metric  stick  for  reference. 


FOU 

R  INCHES  IN  SIX! 

PEENTHS 

,  OF  AN  INCH 

Illllll  III  III 

mm: 

III     III 

l|i|Mi 

l|l|iM 

l|l|l|l 

i|i|)!l 

1 

2 

3 

4- 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

iliiliiil 

lllllllll, 

JJillllll 

lllllllll 

lllllllll 

lllllllll 

iiuiuiJ 

ULlllll 

lllllllll 

lllllllll 

one  decimeter  in  millimeters 

Equivalents. 

1  meter  =  39.37  inches. 

1  decimeter    =  3.937  inches. 
1  centimeter  =  .3937  inch. 
1  inch  =  2.54  centimeters. 

1  mile  =  1.6093  kilometers. 

1  kilometer    =  .6214  of  a  mile. 


254  SCHOOL  ARITHMETIC. 

375.  In  surface  measures  the  principal  unit  is  the  Square 
Meter. 

Table  of  Square  Measure. 

100  square  millimeters  (qmm.)  =  1  square  centimeter  (qcm.). 
100  square  centimeters  =  1  square  decimeter  (qdm.). 

100  square  decimeters  =  1  square  meter  (qm.). 

100  square  meters  =  1  square  dekameter  (qDm.). 

100  square  dekameters  =  1  square  hektometer  (qHm. ). 

100  square  hektometers  =  1  square  kilometer  (qKm.). 

Note.— The  square  dekameter  is  also  called  an  are  (a.),  pronounced 
like  the  word  air,  and  the  square  hektometer  is  called  a  hektare  (ha.). 
The  square  meter  is  sometimes  called  a  centare  (ca.).  These  are  used  in 
measuring  land.  The  area  of  a  farm  is  expressed  in  hektares  ;  that  of  a 
country  in  square  kilometers. 

Equivalents. 

1  square  inch  =  6.452  sq.  centimeters. 

1  square  foot  —  .0929  sq.  meter. 

1  square  yard  =  .8361  sq.  meter. 

1  square  mile  =  2.59  sq.  kilometers. 

1  acre  =  .4047  hektare. 

1  square  meter  —  1.196  sq.  yards. 

1  hektare  =  2.471  acres. 

376.  In  measures  of  volume  theprincipalunit  is  the  Cubic 
Meter. 

Table  of  Cubic  Measure. 

1000  cubic  millimeters  (cmm).  —  1  cubic  centimeter  (ccm.). 
1000  cubic  centimeters  =  1  cubic  decimeter  (cdm.). 

1000  cubic  decimeters  =  1  cubic  meter  (cu  m.). 

Note. — When  used  in  measuring  wood,  the  cubic  meter  is  called  a 
stere  (st.),  pronounced  steer. 


THE   METKIC  SYSTEM.  255 

EqU  IV  ALEUTS. 

1  cubic  inch      =16.387  cubic  centimeters. 
1  cubic  foot      =  .02832  cubic  meter. 
1  cubic  yard      =  .7045  cubic  meter. 
1  cubic  meter  =  1.308  cubic  yards. 

377.  The  principal  unit  of  weiglit  is  the  Gram.     It  is  the 

weight  of  a  cubic  centimeter  of  distilled  water  at  its  maximum 
density. 

Table  of  Weight. 

10  milligrams  (mg.)  =  1  centigram  (eg.). 

10  centigrams  =  1  decigram  (dg.). 

10  decigrams  =  1  gram  (g.). 

10  grams  =  1  dekagram  (Dg.). 

10  dekagrams  =  1  hektogram  (Hg.). 

10  hektograms  =  1  kilogram  (Kg.). 

1000  kilograms  =  1  metric  ton  (T.). 


A  quintal  (Q.)  =  100,000  grams. 
A  myriagram    =  10,000  grams. 

Note. — The  metric  ton  is  the  weight  of  a  cubic  meter  of  water  ;  the 
kilogram  of  a  cubic  decimeter  or  a  liter  of  water,  which  is  about  2.2  lb. 
The  kilogram  is  sometimes  called  a  kilo,  and  is  the  unit  used  in  weighing 
ordinary  articles. 

Equivalents. 

1  pound  avoir.  =r  .4536  kilo. 

1  pound  troy     =  .3732  kilo. 

1  ton  avoir.        =  .9072  metric  ton. 

A  gram  =  15.432  grains. 

378.  The  principal  unit  of  capacity  is  the  Liter  {lee'ter). 
It  is  the  capacity  of  a  cube  whose  edge  is  .1  of  a  meter. 


256  SCHOOL  ARTIHMETIC. 

Table  of  Capacity. 

10  milliliters  (ml.)  =  1  centiliter  (cl.). 
10  centiliters  =  1  deciliter  (dl.). 

10  deciliters  =  1  liter  (1.). 

10  liters  =  1  dekaliter  (DL). 

10  dekaliters  =  1  hektoliter  (HI.). 

10  hektoliters  =  1  kiloliter  (Kl.). 

Note. — The  hektoliter  is  used  in  measuring  grain,  vegetables,  etc. ; 
the  liter  in  measuring  liquids  and  small  fruits. 

Equivalents. 

1  gallon         =  3.786  liters. 
1  bushel        =  .3524  hektoliter. 
1  liter  =  1.0567  liquid  qt. 

1  hektoliter  =  2f  bushels,  nearly. 

ORAL    EXERCISES. 

379.  1.  How  are  the  decimal  values  of  the  principal  units 
named  ? 

2.  Multiples  of  the  principal  units  use  what  prefixes  before 
the  name  of  the  unit  ? 

3.  What  is  the  prefix  which  means  10  ?  100  ?  1000  ? 
.1  ?     .01?     .001? 

4.  Can  you  mention  any  advantages  in  the  use  of  a  metric 
system  of  weights  and  measures  ? 

5.  Since  metric  measures  have  a  decimal  scale,  how  may 
units  of  one  denomination  be  reduced  to  another  ?  How 
should  they  be  written  before  adding  or  subtracting  ? 

6.  How  many  cm.  long  is  this  book  ?     Your  desk  ? 

7.  Cut  a  qdm.  out  of  paper,  and  draw  a  sq.  ft.  How 
many  of  the  former  does  the  latter  equal  ? 

8.  How  many  mm.  in  5  Km.?  In  1  Hm.?  How  many 
cm.  in  25  m.? 

9.  How  many  qcm.  in  a  qm.  ?  How  many  cmm.  in  a  ccm.  ? 
In  a  liter  ? 


THE  METRIC  SYSTEM.  257 

10.  Explain  why  in  measures  of  surface  each  unit  is  100 
times  as  hirge  as  the  next  smaller  unit.  Why  1000  times  as 
large  in  measures  of  volume. 

11.  What  part  of  a  square  meter  is  a  square  decimeter  ? 

12.  For  what  do  we  use  the  inch  ?  The  yard  ?  The 
mile  ?  For  what  are  the  mm.,  the  cm.,  the  m.,  and  the  Km. 
respectively  used  ? 

13.  For  what  purposes  is  the  liter  used  ? 

14.  What  part  of  a  liter  is  100  g.  of  water  ? 

15.  What  is  the  weight  of  a  cubic  meter  of  water  ? 

16.  If  a  train  travels  at  the  ratQ  of  20  m.  a  second,  what  is 
the  rate  in  Km.  an  hour  ? 

WRITTEN    EXERCISES. 

380.  1.  What  part  of  a  cubic  meter  is  a  cubic  foot  ? 

2.  A  girFs  hoop  is  6  m.  in  circumference.  How  many 
times  will  it  turn  in  rolling  a  distance  of  1.08  Km.? 

3.  How  many  kilograms  does  a  barrel  of  flour  weigh  ? 

4.  How  many  square  meters  in  a  circle  whose  diameter  is 
15  meters  ? 

5.  If  sulphuric  acid  is  1.84  times  as  heavy  as  water,  what 
is  the  weight  in  dekagrams  of  26  1.  of  the  acid  ? 

6.  What  is  the  area  of  Virginia  in  square  kilometers  ? 

7.  At  16  cents  a  liter,  what  is  the  cost  of  52.4  HI.  of  olive 
oil  ? 

8.  How  many  kilos  will  a  hektoliter  of  water  weigh  ? 

9.  Mt.  Blanc  is  4800  m.  high.     How  many  feet  high  is  it  ? 

10.  Find  the  weight  in  kilograms  of  a  cubic  foot  of  gold, 
if  gold  is  19.5  times  as  heavy  as  water. 

11.  A  liter  of  mercury  weighs  13.596  Kg.  How  many 
cu  m.  of  mercury  weigh  1  g.  ? 

12.  If  marble  is  2.7  times  as  heavy  as  water,  what  is  the 
weight  of  a  block  2  m.  by  1  m.  by  1  m.? 

13.  What  will  be  the  duty  on  150  liters  of  wine  at  50^  a 
gallon  ? 

17 


^58  SCHOOL  ARITHMETIC. 

14.  A  lot  of  land  containing  62.5  ares  is  sold  for  25  cents 
a  square  meter.     For  how  much  does  the  lot  sell  ? 

16.  The  capacity  of  a  tank  is  60  cu  m.  How  long  will  it 
take  a  pipe  to  fill  it  at  the  rate  of  3.8  Dl.  a  minute  ? 

16.  How  many  steps  2.5  ft.  long  will  a  man  take  in  walk- 
ing a  kilometer  ? 

17.  A  silver  five-franc  piece  weighs  25  g.,  and  is  composed 
of  9  parts  of  pure  silver  and  1  part  of  pure  copper.  AVhat  is 
the  total  weight  of  silver  in  200  five-franc  pieces  ? 

18.  My  cistern  is  2  m.  5  dm.  long,  2^  m.  wide,  and  2  m. 
deep.     How  many  gallons  of  water  will  it  hold  ? 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

381.  1.  If  a  rainfall  is  700  liters  per  hectare,  it  is  how 
many  inches  an  acre  ? 

2.  If  a  ream  of  paper  is  .7872  Dm.  in  thickness,  what  is 
the  thickness  in  millimeters  of  a  single  sheet  ? 

3.  At  $7  a  metric  ton  for  coal,  what  will  the  coal  for  a 
week  cost  if  30  kilos  are  burned  each  day  ? 

4.  Air  being  .001276  as  heavy  as  an  equal  volume  of  water, 
what  is  the  weight  of  the  air  in  a  room  6  m.  long,  4  m.  wide, 
and  3  m.  high  ? 

5.  A  certain  vessel  when  empty  weighs  2.7  Kg.,  and  when 
full  of  water  weighs  4235  Dg.  What  does  it  weigh  when  full 
of  petroleum,  which  is  .7  as  heavy  as  an  equal  volume  of  water  ? 

6.  If  it  costs  $5  to  travel  384  Km.  by  rail,  what  is  the  rate 
of  fare  in  cents  per  mile  ? 

7.  A  hektoliter  of  potatoes  to  the  are  is  equivalent  to  how 
many  bushels  to  the  acre  ? 

8.  What  is  the  distance  in  miles  around  the  earth  through 
the  poles,  if  the  distance  from  the  equator  to  the  pole  is 
10,001,887  meters  ? 

9.  The  distance  between  two  places  on  a  map  is  12.5  centi- 
meters. What  is  the  actual  distance  in  miles,  if  the  scale  of 
the  map  is  1  to  6000  ? 


GENERAL    REVIEW   WORK. 

ORAL    EXERCISES. 

382.  1.  A  man  built  11  rods  of  fence  in  3  days.  How 
much  did  he  build  in  2  days  ? 

2.  How  much  will  a  dozen  lemons  cost,  at  the  rate,  of  2  for 
5  cents  ? 

3.  If  4  apples  cost  5  cents,  how  many  apples  can  be  bought 
for  a  quarter  ? 

4.  Bought  eggs  at  the  rate  of  2  for  3  cents,  and  sold  them 
at  the  rate  of  2  for  5  cents,  gaining  66  cents.  How  many 
dozen  did  I  buy  ? 

5.  A  lady  gave  ^  of  lier  money  for  a  shawl,  and  ^  of  it  for 
a  dress,  and  had  $5  left.     How  much  had  she  at  first  ? 

6.  How  many  pigs  can  be  bought  for  130  when  6  pigs  cost 
$20  ? 

7.  If  7  bats  cost  $3.50,  what  will  a  dozen  cost  at  the  same 
rate  ? 

8.  I  gave  136  for  shoes,  at  the  rate  of  $9  for  3  pairs.  How 
many  pairs  did  I  get  ? 

9.  How  many  books  can  be  bought  for  $12.50,  at  the  rate 
of  3  books  for  $3.75? 

10.  If  f  of  a  barrel  of  sugar  costs  $8,  what  will  5  barrels 
cost  ? 

11.  If  I  of  the  pupils  in  a  school  are  girls,  and  there  are  20 
boys,  how  many  pupils  in  the  school  ? 

12.  A  is  30  years  old,  and  f  of  his  age  is  |  of  his  wife's 
age.     How  old  is  his  wife  ? 

13.  B  has  8|  acres  of  land,  and  A  has  ^  as  many.  How 
many  acres  have  both  ? 


260  SCHOOL  ARITHMETIC. 

14.  If  12  cows  are  worth  3  horses,  and  5  horses  are  worth 
10  yoke  of  oxen,  how  many  oxen  are  worth  4  cows  ? 

15.  If  4  bbl.  of  flour  cost  $14.40,  what  will  }  of  a  barrel 
cost? 

16.  Tom  spent  f  of  his  money  for  a  cap,  and  with  the  re- 
mainder bought  a  dozen  apples  at  the  rate  of  2  for  five  cents. 
How  much  money  had  he  at  first  ? 

17.  If  a  10-foot  pole  casts  a  shadow  18  feet  long,  what  is 
the  length  of  a  pole  that  casts  a  shadow  72  feet  long  ? 

18.  D  has  $50  more  than  E,  and  $30  is  |  of  what  they  both 
have.     How  much  has  E  ? 

19.  One  half  of  a  certain  number  is  12  more  than  ^  of  it. 
What  is  the  number  ? 

20.  Two  thirds  of  A's  money  equals  f  of  B's,  and  both  have 
$8.80.     How  much  has  B  ? 

21.  How  many  square  inches  in  one  side  of  a  two-foot 
cube  ? 

22.  If  5  men  can  build  a  wall  in  Gf  days,  how  long  would 
it  take  11  men  ? 

23.  Fifteen  men  can  build  a  house  in  10  days.  How  many 
men  can  build  it  in  3^  days  ? 

24.  A  can  cut  a  field  of  grass  in  6  days,  B  in  8  days,  and  C 
in  12  days.     In  what  time  can  they  together  cut  it  ? 

25.  A  man  bought  2  hats  for  $5,  and  one  cost  f  as  much 
as  the  other.     What  was  the  cost  of  each  ? 

26.  A  suit  of  clothes  cost  $35.  The  pants  cost  ^  as  much 
as  the  coat,  and  the  vest  ^  as  much  as  the  coat.  What  was 
the  cost  of  each  ? 

27.  Two  fifths  of  Kobert's  money  equals  f  of  Andrew's, 
and  Andrew  has  $2  more  than  Robert.  How  much  has  each  ? 

28.  Two  men  can  husk  a  field  of  corn  in  6  days.  If  one 
of  them  alone  can  husk  it  in  10  days,  how  long  would  it  take 
the  other  ? 

29.  What  will  be  the  cost  of  painting  a  sign-board  20  feet 
long  and  9  feet  wide,  at  the  rate  of  3  square  yards  for  $1  ? 


REVIEW   WORK.  261 

30.  By  selling  a  cow  for  $45  I  gained  as  much  as  she  cost 
me.     Find  the  cost. 

31.  Two  pipes  can  fill  a  cistern  in  8  hours.  If  one  carries 
twice  as  much  water  as  the  other,  in  how  many  hours  can 
each  alone  fill  it  ? 

32.  Harry  has  3  times  as  many  marbles  as  Warren,  and 
Earl  has  5  times  as  many  as  Warren.  If  all  have  30,  how 
many  has  each  ? 

33.  A  man  having  a  lot  8  rods  square,  divided  it  into  4 
equal  lots.  After  selling  one  of  the  lots,  what  part  of  an 
acre  did  he  have  left  ? 

34.  A  is  20  yards  behind  B,  and  runs  9  yards  while  B  runs 
8.     How  far  will  B  run  before  he  is  overtaken  ? 

35.  By  selling  a  stove  for  $34,  a  merchant  gained  ^V  ^^ 
what  it  cost.     How  much  did  it  cost  ? 

36.  Mr.  S  has  20  lots,  each  8  rods  long  and  2  rods  wide. 
How  many  acres  has  he  ? 

37.  If  36  men  can  earn  $a  in  6  days,  how  many  men  can 
earn  $Sa  in  24  days  ?  . 

38.  The  sum  of  two  numbers  is  25,  and  their  difEerence  is 
13.     What  are  the  numbers  ? 

WRITTEN    EXERCISES. 

383.  1.  Sold  42  horses  for  $5600,  thereby  gaining  $13.33^ 
on  each.     What  was  the  cost  a  head  ? 

2.  If  a  horse  can  trot  64  rods  in  half  a  minute,  in  what 
time  can  he  trot  2^  miles  ? 

3.  A  telegraph  line  is  200  miles  long.  If  the  poles  are  150 
feet  apart,  what  is  their  value  at  $1.33^  each  ? 

4.  How  many  boards  14  ft.  long  and  15  in.  wide  would  be 
required  to  cover  the  sides  of  a  shed  28  ft.  long,  21  ft.  wide, 
and  10  ft.  high  ? 

5.  A  street  one  fourth  of  a  mile  long  and  48  feet  wide  is 
to  be  graded  down  half  a  yard.  What  will  be  the  cost  of 
excavating  at  $.12^  a  cubic  yard  ? 


262  SCHOOL  ARITHMETIC. 

6.  How  many  boards  12  ft.  long,  1  ft.  wide,  and  one  inch 
thick,  can  be  made  from  a  square  log  whose  length  is  24  feet, 
and  whose  ends  are  2  ft.  square  ? 

7.  A  cistern  is  22  ft.  long,  14  ft.  wide,  and  8  ft.  deep. 
How  many  gallons  in  it  when  it  is  |  full  ? 

8.  A  log  2  ft.  square  is  24  ft.  long.  How  many  planks 
12  ft.  long,  12  in.  wide,  and  2  in.  thick  can  be  sawed  from 
it? 

9.  An  encyclopedia  averages  764  pages  to  the  volume,  and 
126  lines  to  the  page.  If  the  entire  work  contains  1443360 
lines,  how  many  volumes  are  there  ? 

10.  How  many  gold  dollars  weigh  as  much  as  a  silver  dol- 
lar ? 

11.  A  silversmith  paid  $.60  an  ounce  for  5  lb.  of  silver, 
and  made  it  into  chains  weighing  1  oz.  4  pwt.  each,  which 
he  sold  at  $1.50  apiece.  How  much  did  he  receive  for  his 
labor  ? 

12.  When  it  is  9  a.m.  on  the  meridian  of  Greenwich,  what 
is  the  time  on  the  180th  meridian  ? 

13.  Divide  the  product  of  3^^  and  7^  by  their  sum. 

14.  A  druggist  bought  a  pound  of  calomel  for  $6,  and  sold 
it  at  the  rate  of  5  grains  for  a  cent.     What  was  his  profit  ? 

15.  440  lb.  of  copper  was  made  into  wire,  a  yard  of  which 
weighed  4  oz.     What  was  the  length  of  the  wire  ? 

16.  If  a  cubic  foot  of  granite  weighs  250  lb.,  what  is  the 
weight,  in  tons,  of  a  block  6  ft.  long,  4  ft.  wide,  and  3  ft. 
thick  ? 

17.  If  steel  rails  weigh  180  lb.  per  yard,  how  many  tons 
will  be  required  to  lay  2  miles  of  railway,  one  of  which  has  a 
double  track  ? 

18.  If  a  wheel  is  4  ft.  in  diameter,  how  many  revolutions 
will  it  make  in  going  a  mile  ? 

19.  How  many  horses  can  be  supplied  with  shoes  from  10 
lb.  of  iron,  if  8  oz.  make  one  shoe  ? 

go,  The  distance  over  a  hill  is  60  rods,  and  the  distance 


REVIEW  WORK.  263 

through  on  a  level  is  40  rods.  If  81  posts  are  required  for  a 
fence  from  one  side  to  the  other  on  the  level,  how  many 
would  be  needed  to  build  a  fence  over  the  hill  ? 

21.  Explain  the  effect  of  removing  the  cipher  in  each  of 
the  following  :  750,  025,  .250,  .025. 

22.  llow  often  can  a  3-bushel  bag  be  filled  from  a  bin  con- 
tainiDg  181  bu.  16  qt.  ? 

23.  A  man  bought  500  fence-boards,  each  16  ft.  long  and 
6  in.  wide,  at  $14.50  per  M,  and  50  posts  at  a  quarter  apiece. 
Find  the  total  cost. 

24.  How  many  perches  in  a  pile  of  stone  45  ft.  long,  30 
ft.  wide,  and  4  ft.  high  ? 

25.  A  rectangular  solid  standing  on  a  base  6  in.  square  is 
5J  ft.  high.     How  many  cubic  feet  does  it  contain  ? 

26.  A  trough  4  ft.  in  length  and  2  ft.  square  is  full  of 
water.  What  is  the  weight  of  the  water,  if  a  cubic  foot  of  it 
weighs  1000  ounces  ? 

27.  A  wheel  of  a  bicycle  travels  235.62  yd.  in  making  50 
revolutions.     Wlnit  is  the  radius  ? 

28.  The  area  of  a  triangular  field  is  9  A.  65  sq.  rd.,  and 
the  length  of  its  base  is  70  rods.     What  is  the  altitude  ? 

29.  A  floor  is  24  feet  wide  at  one  end,  16  ft.  at  the  other, 
and  its  area  is  40  sq.  rd.     What  is  the  length  ? 

30.  A  roof  is  50  ft.  long  and  20  ft.  wide  on  each  side. 
What  will  be  the  cost  of  roofing  it  at  $8.75  per  hundred  sq. 
ft.? 


31.  How  many  yards  of  lining  |  of  a  yard  wide  will  be 
required  to  line  5  coats,  each  containing  4|  yards  of  material 
1}  yd.  wide  ? 

32.  Willie  lost  y^  of  his  marbles  less  15  ;  he  then  gave  f  of 
the  remainder  and  8  more  to  John  ;  he  had  32  remaining. 
How  many  had  he  at  first  ? 

33.  The  point  of  a  minute  hand  moves  4  inches  in  two 
Jipurs.     What  is  its  length  ? 


264  SCHOOL   ARITHMETIC. 

34.  Mrs.  Brown  sold  ^g-  of  her  turkeys  less  17  ;  |  of  the 
remainder  less  3,  and  then  had  39  remaining.  How  many 
had  she  at  first  ? 

35.  How  many  quart,  pint,  half  pint,  and  gill  bottles,  of 
each  an  equal  number,  can  be  filled  from  a  vessel  containing 
5  gal.  2  qt.  1  pt.  ? 

36.  I  have  a  lot  the  length  of  whose  sides  is  56,  84,  98,  and 
112  ft.  respectively.  I  desire  to  enclose  it  with  a  fence  four 
boards  high,  using  boards  14  ft.  long.  How  many  will  it 
take  ? 

37.  A  passenger  train  65  yd.  long,  and  running  at  the  rate 
of  I  of  a  mile  a  minute,  met  a  freight  train  moving  ^  as  fast. 
They  passed  each  other  in  5  seconds.  How  long  was  the 
freight  train  ? 

38.  If  9  men  can  do  as  much  work  as  15  women,  and  70 
women  do  as  much  as  25  boys,  and  12  boys  do  as  much  as  36 
girls,  how  many  men  would  it  take  to  do  the  work  of  50 
girls  ? 

39.  Find  the  difference  between  the  largest  fractional  unit 
in  decimals  and  the  largest  fractional  unit  in  common  frac- 
tions. 

40.  If  the  strips  are  laid  lengthwise,  how  many  yards  of 
carpet  27  inches  wide  must  be  bought  for  a  room  18  feet  long 
and  16  feet  wide  ? 

41.  Five  minutes  after  two  ships  pass  each  other  the 
distance  between  them  is  2160  rods.  One  of  them  sails  at 
the  rate  of  35  miles  an  hour.  What  is  the  hourly  speed  of 
the  other  ? 

42.  A  bin  is  25  ft.  long  and  20  ft.  wide.  The  oats  in  the 
bin  is  an  inch  deep,  and  a  bushel  of  it  weighs  32  pounds. 
How  long  will  it  last  5  horses,  if  one  horse  eats  18  pounds  a 
day? 

43.  From  11  a.m.  to  1.30  p.m.  my  watch  gained  10  seconds. 
In  how  many  days  did  it  gain  10  minutes  ? 


REVIEW   WORK.  265 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

384.  1.  When  a  certain  number  is  divided  by  7,  the  quo- 
tient is  equal  to  tlie  ratio  of  5  to  2.     What  is  the  number  ? 

2.  If  ^  of  an  acre  is  worth  f  of  $100,  how  many  thirds  of 
an  acre  can  be  bought  for  f  of  175  ? 

3.  How  many  fractional  units  in  .375  ?  Give  three 
answers. 

4.  If  8  yards  of  cloth  1^  yd.  wide  will  make  Lucy  a  dress, 
how  many  yards  of  48-inch  goods  will  make  her  a  dress  ? 

5.  How  many  fractional  units  equal  to  .25  are  there  in  2|  ? 

6.  In  an  orchard  there  are  18  rows  of  trees.  Between 
every  two  rows  of  trees  there  are  8  rows  of  potatoes.  If 
the  average  yield  of  a  row  is  12  bushels,  what  is  the  value 
of  the  potato  crop  at  $.75  a  bushel  ? 

7.  Is  I  a  fraction,  or  a  number,  or  a  unit  ?  May  it  be  all 
three  ?     Why  ? 

8.  If  marble  weighs  2.8  times  as  much  as  water,  bulk  for 
bulk,  what  is  the  weight  of  a  block  of  marble  12  ft.  9  in. 
long,  4.5  ft.  wide,  and  3.2  ft.  thick  ? 

9.  The  average  depth  of  a  certain  rainfall  was  .25  of  an 
inch.  What  weight  of  water  fell  on  .a  lot  40  ft.  by  60  ft.,  if 
1000  oz.  of  water  measures  a  cubic  foot  ? 

10.  If  a  train  loses  f  of  an  hour  in  running  80  miles  at  18 
miles  an  hour,  in  how  many  hours  does  it  run  360  miles 
when  running  at  the  regular  speed  ? 

11.  If  4  horses  can  draw  80  bushels  of  wheat,  00  lb.  to 
the  bushel,  on  a  wagon  whose  weight  is  000  lb.,  how  many 
bushels  can  2  horses  draw  on  a  wagon  that  weighs  600  lb.  ? 

12.  Does  the  expression  .^  mean  anything  ?     If  so,  what  ? 

13.  A  cubical  box  is  4.8  m.  on  an  edge.  How  many 
hektoliters  of  oats  will  it  hold  ?     How  many  bushels  ? 

14.  How  many  bricks  20  cm.  long  and  10  cm.  wide  will  it 
take  to  pave  a  sidewalk  2.4.m.  wide  and  1.4  Km.  long  ? 

15.  At  43^  a  cubic  meter,  what  will  it  cost  to  macadamize 
a  road  1  Km.  long  and  7  m.  wide,  to  the  depth  of  46  cm.  ? 


PERCENTAGE. 

385.  1.  Warren  had  100  cents  and  spent  1  cent.  What 
part  of  his  money  did  he  spend  ? 

2.  Boyd  had  $1  and  spent  2  cents.  AVhat  part  of  his  dollar 
did  he  spend  ? 

3.  Emma  misses  5  words  in  a  hundred.  What  part  does 
she  miss  ? 

'4.  A  man   had  400  sheep,  and   10  of  every  hundred   were 
killed  by  dogs.     How  many  were  killed  ? 

6.  A  farmer  having  50  hogs  lost  10  hundredths,  or  10  per 
cent  of  them.     How  many  did  he  lose  ? 

6.  If  yJ^,  or  7  per  cent,  of  the  pupils  in  a  school  of  100 
are  absent,  how  many  are  absent  ?  How  many  are-  present  ? 
How  many  per  cent  are  present  ? 

7.  WhAt  is  6  per  cent,  or  .06,  of  1100  ?  Of  1300  ?     Of  1500  ? 

386.  Per  cent  means  hundredths. 

Thus,  5  per  cent  of  a  number  means  5  hundredths  of  it. 
Note. — The  phrase  "  per  cent  "  is  from  the  Latin  per  centum,  by  the 
hundred. 

387.  The  symbol  ^  stands  for  the  words  per  cent,  and 
means  either  per  cent  or  hundredths. 

Thus,  %%  is  .06,  and  is  read  6  per  cent  or  6  hundredths. 

1.  How  do  we  express  cents  ?     Hundredths  ? 

2.  Since  per  cent  is  so  many  hundredths,  how  may  we 
express  per  cent  9 

3.  Is  there  any  difference  in  value  between  20^,  .20,  and^  ? 

4.  Express  VZ^  per  cent  in  three  ways.  ^  per  cent  in  3 
ways. 

$.  Explain  how  5  per  cent,  or  5^  =  .05,  or  -^-q,  or  ^. 


PERCENTAGE. 


267 


6.  Express  225  per  cent  in  two  ways  ;  90  per  cent ;  100  per 
cent. 


388.  Change  to  per  cent : 


1.  .06  = 

2.  .15  = 

3.  .25  = 

6^. 

7.  Th     = 

8.  ^\     = 

9.  m     = 

7^. 

13.  i  = 

14.  f  = 

15.  i  = 

50^. 

19. 
20. 
21. 

.OOi  = 
.00|  =f 
.005  = 

4.  .39  = 

10-  2  = 

16.*  = 

22. 

.0005  = 

5.  1.35  = 

6.  3.   = 

11.  1.375  = 

12.  .01  = 

17.  t  = 

18.  A  = 

23. 
24. 

.2775  = 
.0325  = 

i^. 


25.  226  hnndredtlis  =  liow  many  per  cent  ? 
389.  Change  to  decimal  fractions  : 
1.  6^     =.06    6.  i^  =.005  11.  100^  =1.00  16.  162^^  =  1.625 

17.  237^^  = 


2.  1 
3.2 
4. 

5.  99J^  = 


7.J^  = 

8.  :J^    = 

9.  f^  = 
10.  i|^= 


12.  227^  = 

13.  3000^= 
14.^^^  = 
15.  6i^     = 


18.  266f^  = 

19.  267i?^= 


390.  Change  to  common  fractions  : 
161 


3^^=i    6.  16|^  =100=*  ^-  ^'^^=i  13.  87J^  =1 

35^=  6.  37^^  =  10.  150^=     14.  .283    = 

7.  83^^  =  11.  180^=     15.  .375    = 

8.  116|^=  12.  225^=     16.  233^^  = 


75^  = 


391.  The  following  per  cents  and  their  equivalents  are  so 
often  used  that  pupils  should  be  able  to  give  their  values  in 
the  different  forms  at  sight. 

Drill  Table. 


1  =  100^. 

1=75^. 

i 

= 12J^. 

A 

=  6|! 

i=  50^. 

i  =  20^. 

f 

=  37^^. 

jV 

=  5^ 

i  =  33^^. 

1  =  60^. 

f 

=  >->H^. 

A 

=  4i^ 

1  =  c^Hfc. 

i  =  mfo. 

1 

=  871^. 

A 

=  3^ 

i=  25^. 

i  =  m^- 

1 
IT 

=  10  f. 

iijs 

=  1^ 

268  SCHOOL  ARITHMETIC. 

392.  The  result  obtained  by  taking  a  certain  per  cent 
(meaning  a  stated  number  of  hundredths)  of  a  number  is 
called  the  Percentage. 

The  name  Percentage  is  also  applied  to  that  portion  of  arithmetic 
which  involves  the  taking  of  per  cents. 

393.  The  number  of  which  the  per  cent  is  taken  is  called 
the  Base. 

394.  The  per  cent  taken  is  called  the  Rate. 

Thus,  in  6^  of  50  =  3,  the  Q%  is  the  rate.  The  6  alone  is  usually 
called  the  rate  per  cent. 

395.  Using  ih^  first  letters  of  the  vford.^ percentage,  base, 
and  rate  to  represent  the  numbers  called  by  these  names,  we 
readily  express  in  the  form  of  equations  the  relations  that 
these  numbers  bear  to  each  other. 

From  the  definition  of  percentage, 

p  =  hr.  (1) 

Dividing  both  members  of  (1)  by  b, 

|  =  r,  orr=f.  (2) 

Dividing  both  members  of  (1)  by  r, 

6  =  1.  (3) 

r  ^  ' 

In  (2)  we  have  a  'product  {percentage)  and  one  of  the  factors  (base)  to 
find  the  other /ac/or  (ra^e).  What  have  we  in  (1)  ?  In  (3)  ?  What  rela- 
tion, then,  do  the  base,  rate,  and  percentage  bear  to  each  other  ? 

396.  One  hundred  per  cent  of  a  number  is  the  number 
itself. 

Thus,  100^  of  50  is  50. 

397.  To  fiud  a  given  per  cent  of  any  number. 

1.  What  part  of  a  number  is  10  hundredths  of  it  ?  10^ 
of  it? 


PERCENTAGE.  269 

2.  How  many  hundredths  in  1  ?  How  many  cents  in  $1  ? 
How  many  per  cent  in  1  ? 

3.  What  per  cent  of  a  number  is  ^^  of  it  ?  J  of  it  ?  All 
of  it? 

4.  Since  Ifo  of  a  number  =  j^^  of  it,  10^  =  -^jPq  or  Vtt  of  it, 
and  50^  =  ^%-  or  ^  of  it,  what  does  100^  equal  ? 

6.  Since  any  number  is  100^  of  itself,  and  the  base  is  the 
number  of  which  the  per  cent  is  taken,  the  base  equals  what 
per  cent  ? 

6.  If  you  lose  4^  of  your  money,  what  per  cent  do  you 
have  left  ?     100^  -  4^  =  (     ). 

What  is  : 

7.  1^  of  50  ?  13.  5^  of  50  ? 

8.  10^  of  50^  ?  14.  100^  of  50  ? 

9.  12^  of  40  ?  15.  50^  of  100  sheep  ? 

10.  161^  of  66  ?  16.  75^  of  120  horses  ? 

11.  33J^  of  60  ?  17.  83i^  of  24  quarts  ? 

12.  66f^  of  72  ?  18.  100^  of  100  bushels  ? 

WRITTEN    EXERCISES. 

398.  1.  Find  20^  of  I960. 

(a)  (b) 

.20^  =  .20  or  i  100^  of  $960  =  m  of  $960. 

.20  of  $960  =  $192.  .-.  1%  of  $960  =  yio  of  $960,  or  $9.60, 

Or  and  20^  of  $960  =  20  x  $9.60,  or  $192. 
i  of  $960  =  $192. 

Have  the  pupil  solve  the  above  problem  by  substituting 
in  the  equation 

p  =  ir. 

2.  Find  j^  of  $800.      - 

(a)  (b) 

100$^  of  $800  =  $800.  1%  of  $800  =  $8.00. 

.-.  Ifc  of  $800  =  T^TT  of  $800,  or  $8,  1%  of  $800  =  f  x  $8.00,or  $6. 

and  i%  of  $800  =  ^  x  $8,  or  $6. 


2Y0  SCHOOL  ARITHMETIC. 

Suggestion. -=-The   pupil  should  be  tauglit  to  select  and  apply  the 
method  that  is  most  convenient  in  each  particular  problem. 


3. 

12fc  of  $240.50. 

7.  325^  of  55.2  rods. 

4. 

7bfo  of  11286.45. 

8.  133^  of  17824. 

6. 

16f^  of  120  sheep. 

9.  90 fo  of  .0577. 

6. 

1^  of  1200. 

10.  li^of^V 

11.  What  is  tlie  difference  between  ofo  of  1120,  and  120^ 
of  $5? 

Query. — Is  the  percentage  a  factor  or  a  product  ? 

12.  How  much  had  I  left  after  paying  out  15^  of  my 
$3000  ? 

13.  The  owner  of  a  threshing  machine  charges  2^^  for 
threshing  a  crop  of  275  bushels.     How  much  does  he  get  ? 

14.  A  man  who  owed  $1750  was  able  to  pay  only  39  per 
cent.     How  many  dollars  could  he  pay  ? 

15.  If  a  foot  of  rope  shrinks  4|^^  when  wet,  how  much 
would  500  feet  of  rope  shrink  ? 

16.  If  A^s  income  is  $1000  a  year  and  he  saves  25^  of  it, 
how  much  Avill  he  save  in  25  years  ? 

17.  I  bought  40  head  of  cattle  for  $166()|,  and  sold  them 
at  a  profit  of  3^^.     What  did  I  make  ? 

18.  Dickson  and  Tribby  engaged  in  business,  each  with 
$1250.  Tribby  gained  33|^^  of  his  capital,  and  Dickson  37^^ 
of  his  capital.  How  much  did  Dickson  gain  more  than 
Tribby  ? 

19.  A  farmer  raises  500  bushels  of  grain,  of  which  29^ 
was  wheat,  47^  rye,  and  22^  oats.  How  many  bushels  of 
each  did  he  raise  ? 

20.  I  own  }  of  a  mill  and  sell  33^^  of  my  share.  What  part 
of  the  mill  do  I  sell  ? 

21.  A  barrel  that  will  hold  42  gallons  is  66|^  full.  How 
many  gallons  does  it  contain  ? 

22.  A  man  has  $1500.  He  spends  6Gf^  of  it,  and  gives 
away  6^  as  much  as  he  spends.     How  much  has  he  left  ? 


PERCENTAGE.  271 

ft3.  If  pure  air  consists  of  20.0265^  of  oxygen  gas  and 
79.9735^  of  nitrogen,  how  much  oxygen  in  1500  cu.  ft.  of 
air  ?     How  much  nitrogen  ? 

24.  If  25^  of  a  certain  ore  is  melted,  and  If^  of  the  metal 
is  silver,  how  much  silver  in  a  ton  of  the  ore  ? 

25.  A  certain  lot  of  cane  has  89^^  juice,  and  the  juice  con- 
tains 11.4^  sucrose.     How  much  sucrose  in  5  tons  of  cane  ? 

26.  A  man  has  a  library  of  1600  volumes.  14^  are  biog- 
raphy, 62^  are  history,  and  83^^  of  the  remainder  are  fic- 
tion.    How  many  volumes  of  fiction  in  his  library  ? 

27.  A  maltman  malts  1500  bushels  of  barley,  which  in  the 
process  increases  12|^^.     How  many  bushels  of  malt  has  he  ? 

28.  An  agent  sells  25  bicycles  at  $60  each,  and  is  allowed 
15^  of  the  receipts.     How  much  does  he  make  ? 

29.  Water  is  composed  of  88.9^  of  oxygen  and  11. Ij^  of 
hydrogen.  How  many  pounds  are  there  of  each  in  1  cu.  ft. 
of  water  ? 

399.  To  find  what  per  cent  one  number  is  of 
anotlier. 

1.  12  is  what  part  of  24  ?  How  many  hundredths  of  24  ? 
What  per  cent  of  24  ? 

2.  15  is  what  part  of  45  ?  How  many  hundredths  or  per 
cent  of  45  ? 

3.  $5  is  what  part  of  $25  ?  What  per  cent  of  $25  ?  What 
is  the  ratio  of  $5  to  $25  ? 

What  per  cent  of  : 

4.  24  is  12  ?  8.  $25  is  $10  ?  12.  1  is  J  ? 

5.  7  is  21  ?  9.  30  yd.  is  20  yd.  ?        13.  9  is  f  ? 

6.  8  is  18  ?  10.  100^  is  50^  ?  14.  f  is  i  ? 

7.  30  is  5  ?  11.  5  is  5  ?  16.  f  is  f  ? 

16.  A  boy  having  10  cents  gave  his  sister  5  cents  ?  What 
.per  cent  of  his  money  did  he  give  away  ? 

17.  A  teacher  whose  salary  is  $1250  spends  $1000.  What 
per  cent  of  his  salary  does  he  spend  ? 


272  SCHOOL  ARITHMETIC. 

18.  A's  money  is  twice  B's.     What  per  cent  of  A's  money 
is  B^s  ?     What  per  cent  of  B's  is  A's  ? 


WRITTEN     EXERCISES. 

400. 

1.   9  is  what  per  cent  of  30  ? 

30: 
.-.   1   . 

and  9  : 

(a) 
=  100^  of  30. 
=  ^h  of  100^, 
=  9  X  ^1%,  or 

r 

or  ^H  of  30, 
■  30^  of  30. 

(c) 
=  !=,%  =  .30,  or 

9  is 
and 

30^. 

(b) 
=  .30, 

of  30, 
or  30^. 

2.  If  a  miller  takes  4  qt.  for  toll  from  every  bushel,  what 
per  cent  does  he  take  for  toll  ? 

3.  3^  is  what  per  cent  of  20  ? 

4.  Edward  bought  2  lb.  of  candy.  He  ate  4  oz.  and  gave 
away  ^  lb.     What  per  cent  of  his  candy  did  he  have  left  ? 

5.  From  a  cask  containing  66^  gal.,  26.6  gallons  were 
drawn.     What  per  cent  of  the  whole  remained  in  the  cask  ? 

6.  If  gold  coin  is  9  parts  pure  and  1  part  alloy,  what  per 
cent  is  pure  ? 

7.  An  attorney  charges  $68.75  for  collecting  $550.  What 
per  cent  does  he  charge  ? 

8.  Mr.  S  paid  $45  for  the  use  of  $750.  What  per  cent  did 
he  pay  ? 

9.  What  per  cent  is  a  pound  avoirdupois  of  a  pound  troy  ? 

10.  25^  of  I  of  a  number  is  what  per  cent  of  f  of  it  ? 

11.  A  merchant  buys  5  gross  of  pens,  and  sells  5  dozen. 
What  per  cent  of  them  does  he  sell  ? 

12.  Frank  has  $10  and  Kay  $4.  What  per  cent  of  Ray's 
money  is  equal  to  Frank's,  and  what  per  cent  of  Frank's 
money  equals  Eay's  ? 

13.  What  per  cent  of  his  time  does  a  man  sleep  who  sleeps 
7 J  hr.  out  of  24  ? 

14.  If  7  lb.  of  a  certain  article  lost  4  oz.  by  drying,  what 
per  cent  of  its  original  weight  was  water  ? 


PERCENTAGE.  273 

15.  In  a  mixture  of  copper  and  zinc,  the  copper  is  to  the 
zinc  as  3|-  to  2^.  Express  the  percentage  of  each  ingredient 
in  the  mixture. 

16.  If  carpeting,  which  should  be  one  yard  wide,  is  only  34J 
inches  wide,  what  per  cent  should  be  deducted  from  the  price  ? 

17.  If  I  sell  ^  of  my  interest  in  a  business  to  one  man  and 
i  of  it  to  another,  what  per  cent  have  I  remaining  ? 

18.  If  to  23  gallons  of  alcohol  2  gallons  of  water  are  added, 
what  per  cent  of  the  mixture  is  water  ? 

19.  In  an  examination  50  questions  were  asked,  of  which 
A  answered  45,  B  35,  and  C  18.  What  per  cent  did  each 
make  ? 

20.  If  B's  age  is  33Jj^  more  than  A^s,  A's  age  is  what  per 
cent  less  than  B^s  ? 

21.  If  a  gold  ring  is  18  carats  fine,  what  per  cent  of  it  is 
gold? 

22.  If  a  piece  of  bronze  weighing  7f  pounds  contains  6.5 
pounds  of  copper,  what  per  cent  of  the  bronze  is  copper  ? 

23.  Of  25320  votes  cast  in  a  certain  city,  A  received 
11394  and  B  the  remainder.     What  per  cent  did  B  receive  ? 

24.  An  army  paymaster  receives  $125000,  but  embezzles 
15000  of  it.  What  per  cent  of  the  money  does  the  govern- 
ment lose  ? 

25.  In  a  certain  year  (1898)  501,066,681  passengers  were 
carried  on  railways  in  the  United  States,  and  221  were  killed. 
What  per  cent  were  not  killed  ? 

26.  The  area  of  North  America  is  9,350,000  sq.  mi.,  and 
the  area  of  the  Missouri-Mississippi  basin  is  1,250,000  sq. 
mi.  What  per  cent  of  the  area  of  the  continent  is  drained 
by  these  rivers  ? 

401.  To  find  a  number  ivhen  a  certain  per  cent  of 
it  is  given. 

1.  15  is  3  times  what  ?    |  of  what  ?    .03  of  what  ?    3  per 
cent  of  what  ? 
18 


274  SCHOOL   ARITHMETIC. 

2.  24  is  f  of  what  ?     .06  of  what  ?     6  per  cent  of  what  ? 
Find  the  Dumber  of  which  : 


3.  10  is  10^. 

6.  30  is  12^^. 

9.  f  is  50^. 

4.  60  is  25^. 

7.  $150  is  50^. 

10.  50  is  Ifo. 

5.  36  is  33i^. 

8.  1  is  100^. 

11.  f  is  5^. 

12.  Of  passengers  aboard  a  ship  16f^  of  the  number,  or 
800  persons,  were  lost.  What  was  the  number  of  persons 
aboard  ? 

13.  ^  of  f  of  a  yard  is  20^  of  what  ? 

14.  A  bin  holds  60  bushels  of  wheat,  which  is  3^  of  a 
farmer's  crop.     How  many  bushels  did  he  have  ? 

WRITTEN    EXERCISES. 

402.  1.  30  is  5^  of  what  number  ? 

(a)  (b) 

5^  of  a  number  =  30.  5^,  or  <fo  of  the  number  =  30. 

.-.  \%  of  the  number  =  \  oi  30,  or  6,  \%  of  the  number  = 

and  100^  of  the  number  =  100  x  6,  or  600. 

2.  24  is  f  ^  of  what  number  ? 

1%  of  a  number  =  24. 
.•.  \%  of  the  number  :=  ^  of  24,  or  8. 
.•.  %%,  or  \%,  of  the  number  =  8  x  8,  or  64, 
and  100^  of  the  number  =  100  x  64,  or  6400. 

3.  A  paid  B  $20,  which  was  m  of  what  he  owed  him. 
What  was  the  debt  ? 

4.  A  farmer  sold  560  bushels  of  wheat,  which  was  85^  of 
his  crop.     How  many  bushels  did  he  raise  ? 

6.  A  boy  after  selling  60^  of  his  berries  received  80^  for 
the  remainder  at  10^  a  quart.  How  many  quarts  had  he  at 
first? 

6.  A  charity  concert  realized  $200.85  for  the  benefit  of  the 
poor.  The  expenses  were  35^  of  the  receipts.  W^hat  were 
the  expenses  ? 


.       PERCENTAGE.  275 

7.  A  milkman  sold  milk  at  7  cents  a  quart,  which  was 
233^^  of  the  cost.     What  did  it  cost  a  gallon  ? 

8.  If  80^  of  the  pages  in  Webster's  International  Dictionary 
is  1688,  how  many  pages  does  it  contain  ? 

9.  When  gold  was  worth  20^  more  than  paper  money, 
what  was  the  value  in  gold  of  a  dollar  bill  ? 

10.  A  father  gave  the  contents  of  a  purse  to  his  3  sons. 
To  Charles  he  gave  33^^  of  the  money  ;  to  John,  37^^  of  it ; 
and  to  James,  20^  of  the  money,  or  $2.  How  many  dollars 
were  in  the  purse  at  first  ?. 

11.  A  man  sold  50  cords  of  wood,  losing  thereby  $25,  or 
10^.     What  did  the  wood  cost  him  a  cord  ? 

12.  The  land  surface  of  the  earth  is  about  52,000,000 
square  miles,  which  is  33^^  of  the  water  surface.  What  is 
the  extent  of  the  water  surface  ? 

13.  If  2^  bbl.  of  apples  are  }^  of  the  yield  of  my  orchard 
this  year,  how  many  bushels  did  the  orchard  produce  ? 

14.  By  adding  $695.75  to  my  money  in  bank,  I  increased 
it  20^.     How  much  did  I  have  in  bank  before  the  increase  ? 

15.  A  rug  12  feet  long  and  10  feet  wide  covers  24^  of  the 
floor  of  a  room  25  feet  long.     How  wide  is  the  room  ? 

403.  To  find  a  miinber,  liayiDg  given  another  num- 
ber which  is  a  certain  per  cent  more  or  less  than  the 
required  number. 

1.  What  number  increased  by  5  times  itself  becomes  30  ? 
'  2.  What  number  increased  by  .06  of  itself  becomes  212  ? 
By  6^  of  itself  becomes  212  ? 

3.  What  number  diminished  by  f  of  itself  becomes  30  ? 
By  .06  of  itself  becomes  188  ?  By  6^  of  itself  becomes 
188  ? 

4.  My  salary  is  $75  a  month,  which  is  25^  more  than  it 
was  last  year.     What  was  my  salary  last  year  ? 

5.  I  pay  $12  a  week  for  board  this  year,  which  is  20^  less 
than  I  paid  last  year.     What  did  I  pay  last  year  ? 


276  SCHOOL  ARITHMETIC. 

WRITTEN     EXERCISES. 

404.  1.  What  number  increased  by  15^  of  itself  equals 

1150? 

100^  of  the  number  =  the  number. 
100^  +  15fe,  or  115^  of  the  number  =  1150. 
.-.  Ifc  of  the  number  =  1150  -f-  115,  or  10, 
and  100^  of  the  number  =  100  x  10,  or  1000. 

2.  What  number  diminished  by  90^  of  itself  equals  126  ? 

100^  of  the  number  =  the  number. 
100^  -  90^,  or  10^  of  the  number  =  126. 
.-.  1%  of  the  number  =  /o  of  126,  or  12.6, 
and  100^  of  the  number  =  100  x  12.6,  or  1260. 

3.  A  expends  in  a  week  $24,  which  exceeds  his  earnings 
by  33^^.     What  are  his  earnings  f 

4.  A  mechanic  has  had  his  wages  twice  reduced  10^. 
What  did  he  receive  before  the  reductions  if  he  now  gets 
$2,025  a  day? 

5.  If  f  of  a  number  is  240  more  than  66f  ^  of  it,  what  is 
the  number  ? 

6.  A  school  enrolls  87  boys,  which  is  16^  more  than  the 
number  of  girls.     How  many  pupils  in  the  school  ? 

7.  A  flock  of  sheep  has  been  increased  by  250^  of  its  num- 
ber, and  now  numbers  1050.  What  was  the  original  num- 
ber ? 

8.  I  gave  away -40^  of  my  money,  and  had  $2  left.  How 
much  had  I  at  first  ? 

9.  If  I  double  my  capital,  and  in  addition  make  25^  of  it, 
and  then  have  $22500,  what  was  my  capital  ? 

10.  The  fraction  -^  is  20^  more,  and  20^  less,  than  what 
fractions  ? 

11.  The  Mississippi  is  4200  miles  long,  and  is  5^  longer 
than  the  Nile,  which  is  6|^  longer  than  the  Amazon.  Find 
the  length  of  each  river. 

12.  There  are  48  gallons  in  two  casks,  one  of  which  con- 
tains 40;^  less  than  the  other.     How  much  is  in  each  cask  ? 


PERCENTAGE.  277 

13.  John  spent  80^  of  his  money,  and  had  $400  left.  Hov/ 
much  had  he  at  first  ? 

14.  Tlie  time  past  noon  is  66f  j^  of  the  time  to  midnight. 
What  is  the  time  ? 

15.  A  man  worth  $990,000  is  unhappy  bscause  his  fortune 
is  1^  less  than  his  neighbor's.    What  is  his  neighbor  worth  ? 

16.  The  difference  between  two  numbers  is  425,  and  one  of 
them  is  25^^  greater  than  the  other.  What  are  the  num- 
bers ? 

17.  After  spending  20j^  of  her  money  for  a  coat,  and  80^ 
of  the  remainder  for  a  watch,  a  lady  had  $12  left.  How 
much  had  she  at  first  ? 

18.  A  farmer  gave  40^  of  his  land  to  one  son,  33^^  of 
60^  to  another,  870  of  the  remainder  to  another,  and  the 
remainder,  8  acres,  to  his  daughter.  How  many  acres  had 
he  at  first  ? 

19.  Seventy-five  per  cent  of  a  number  exceeds  |  of  it  by 
60fo  of  7.     What  is  the  number  ? 

Complete  the  following  equations  : 

20.  5^  of  75  =  (     ^)  of  15.  26.  50^  of  2  =  (     ^)  of  200. 

21.  20^  of  35  =  50^  of  (     ).  27.  33^^  of  f  =  12J  ^of  (     ). 

22.  16^  of  (     )  =  4^  of  200.  28.  40^  of  40^  =  24^  of  (     ^). 

23.  (     ^)  of  27  =  27,^  of  33J.  29.  (     ^)  of  .85  =  500^  of  .034. 

24.  (     ^)ofl=i^of2.  30.  i^ofC     )  =  75^off 

25.  75^ofl  =  (     ^)ofJ.  31.  2i^  of  3.6  =  (     ^)  of  .9. 

32.  6^  of  $100  =  8^  of  ($     ). 

33.  (     ^)  of  twice  a  number  =  6^  of  thrice  the  number. 

34.  5^  of  (      times)  a  number  =  40^  of  half  the  number. 

35.  62^^  of  .24  :  .6^  of  50  =  (     ^)  of  25  :  25^  of  10. 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

405.  1.  How  many  pounds  of  tallow  must  be  mixed  with 
8.5  pounds  of  rosin  that  the  mixture  may  contain  15j^  of 
tallow  ? 


278  SCHOOL  ARITHMETIC. 

2.  On  Jan.  1  a  man  weighed  150  lb.  In  that  month  he 
lost  2^^  in  weight,  and  in  February  gained  2^^.  What  per 
cent  of  his  weight  Jan.  1  was  his  weight  on  the  first  day  of 
March  ? 

3.  Sugar  is  composed  of  49.85G^  of  oxygen,  43.265^  of 
carbon,  and  the  remainder  hydrogen.  How  many  pounds  of 
each  in  1  ton  of  sugar  ? 

4.  The  capacity  of  a  bin  9  feet  long,  5  feet  wide,  and  4 
feet  deep  equals  36^  of  how  many  bushels  of  grain  ? 

5.  A  tank,  whose  capacity  is  168  gallons,  discharges  72 
gallons  an  hour,  which  is  25^  less  than  it  receives.  In  what 
time  will  it  be  filled  ? 

6.  A  man  has  his  money  invested  as  follows  :  13000  at  4^, 
$500  at  6^,  and  $1200  at  6^^.  If  he  should  invest  the  whole 
amount  at  bfo,  would  he  gain  or  lose,  and  how  much  ? 

7.  The  earth's  radius  is  6370  Km.,  and  the  sun's  is  10856^ 
of  the  earth's.     Find  the  radius  of  the  sun. 

8.  In  1898  the  number  of  pupils  enrolled  in  the  schools  of 
the  United  States  was  14,652,492,  which  was  20.53,^  of  the 
population.  What  was  the  population  ?  Of  those  enrolled, 
10,089,620  were  in  daily  attendance.  What  was  the  per  cent 
of  attendance  ? 

9.  In  a  certain  year  (1898)  the  gold  and  silver  moneys  in 
this  country  was  $1,354,283,142,  and  the  paper  money  was 
$1,143,946,669,  or  a  total  of  $2,498,229,811.  Theji^er  capita 
circulation  was  $25.19.  If  the  population  was  74,925,000, 
what  per  cent  of  the  total  was  in  circulation  ? 

10.  The  wealth  of  the  United  States  by  the  census  of  1890 
was  65,037  millions  of  dollars.  The  increase  of  wealth  from 
1880  to  1890  was  49^.  At  the  same  rate  of  increase,  what 
wealth  should  the  census  of  1900  show  ? 

11.  In  the  Spanish-American  War  the  deaths  in  the  army 
from  all  causes.  May  1  to  Sept.  30,  1898,  were  2910,  and  the 
total  force  was  274,837  soldiers.  What  was  the  per  cent  of 
loss  by  death  ? 


PERCENTAGE.  279 

12.  Of  the  $1,210,292,097  worth  of  exports  sent  abroad 
from  the  United  States  in  1898,  70.61^  of  the  total  were 
farm  products.  What  was  the  value  of  the  farm  products 
exported  that  year  ? 

13.  In  1890  the  manufacturing  industries  of  the  United 
States  employed  3,599,292  males,  which  was  80.4^  of  the 
total  number  of  employes.  Of  this  total  2.68^  were  children, 
and  16.92^  were  females.  How  many  women  and  children 
were  employed  in  manufactures  ? 

14.  In  the  manufacturing  industries,  the  office  forces  (in- 
cluding firm  members)  iorm  9.78^  of  the  aggregate  em- 
ployes, and  they  receive  17.17^  of  the  total  wages.  There- 
fore an  average  office  employe  receives  how  many  times  as 
much  salary  as  an  operative  ? 

COMMERCIAL.   DISCOUNT. 

406.  1.  The  list  price  of  a  book  is  $1,  but  copies  are 
offered  for  introduction  at  25^  off.  What  is  the  introduction 
price  ? 

2.  A  sewing  machine  sells  for  140  on  3  months  time,  or 
lO^i^  off  for  cash.     What  is  the  cash  price  ? 

3.  A  bill  of  goods  was  sold  at  10^  less  than  list  price, 
which  was  $20.  If  5^  was  deducted  for  cash,  what  was  the 
net  cash  price  ? 

4.  Does  it  make  any  difference  in  the  net  price  whether 
the  discount  is  10^  off  and  5^  for  cash,  or  5^  off  and  10^  for 
cash  ? 

407.  An  allowance  or  reduction  from  a  list  price,  or  from 
the  amount  of  a  bill,  is  called  a  Commercial  Discount. 

Manufacturers,  publishers,  and  wholesale  dealers  usually 
bill  goods  to  the  trade  at  fixed  list  or  catalogue  prices,  subject 
to  discounts. 

1.  As  the  market  varies  they  revise  the  discounts  instead 
of  changing  the  prices. 


280  SCHOOL  AHITHMETIC. 

2.  Many  houses  print  on  their  bill-heads  the  discounts 
allowed  ;  as,  ''  Terms  :  30  days  net,  or  2^  if  paid  in  10  days/' 

408.  When  several  discounts  are  allowed,  one  is  first 
deducted,  then  another  is  computed  on  the  remainder  and 
deducted,  and  so  on  for  each  successive  discount.' 

Thus,  25^  and  b%  means  a  discount  of  25^  from  the  list  price  and 
then  b%  from  the  remainder. 

409.  The  list  price  of  an  article  less  the  discounts  is  the 
net  price  ;  and  the  amount  of  a  bill  less  the  discounts  is  the 
net  amount. 

WRITTEN    EXERCISES. 

410.  1.  A  merchant  buys  goods  listed  at  $1200,  the  dis- 
counts being  25^,  20^,  and  10^.     Find  the  net  price. 

(b) 

$1200  X     .25  =  $300,  first  discount. 
$1200  —  $300  =  $900,  first  remainder. 
$900    X      .20  =  $180,  second  discount. 
$900    —  $180  =  $720,  second  remainder. 
$720    X     .10  =  $72,    third  discount. 
$720    -    $72  =  $648,  thene^ijWce. 
$648 
The  calculation  is  often  conveniently  made  as  in  (a). 

Notes. — 1.  When  several  discounts  are  given,  the  list  price  is  the  hose 
in  computing  the^rs^  discount  only. 

2.  It  is  immaterial  in  what  order  the  discounts  are  taken. 

3.  In  stating  commercial  discounts,  the  sign  %  is  very  often  omitted. 

2.  A  bill  of  goods  amounted  to  $2400.  If  20^  off  was 
allowed,  what  was  paid  for  the  goods  ? 

3.  A  bill  of  $850  was  discounted  8^  and  5^.  What  were 
the  discount  and  the  amount  paid  ? 

4.  What  is  the  cash  price  of  a  carriage  listed  at  $230,  15^ 
off,  and  5^  for  cash  ? 

5.  Find  the  net  amount  of  a  bill  of  $280.50,  the  discounts 
being  15^,  10^,  and  5^. 


4 
5 

(a) 

$1200 
300 

$900 
180 

10 

$720 
72 

PERCENTAGE.  ^81 

6.  How  much  greater  is  a  single  discount  of  40^  on  a  bill 
of  $210,  than  two  successive  discounts  of  30^  and  10^  ? 

7.  Which  is  better,  a  single  discount  of  19^  or  three 
successive  discounts  of  12^,  8^,  and  1^  off  a  bill  of  $647  ? 

8.  A  bill  of  hardware  amounting  to  1375  was  bought  March 
1,  "Terms  :  3  mo.,  or  10^  off,  30  da/'  How  much  money 
paid  the  bill  April  1  ? 

9.  A  school  district  bought  7000  grammars  listed  at  $.60, 
discount  20^.  If  5^  was  allowed  for  cash  payment,  what 
sum  did  the  district  remit  ? 

10.  Goods  listed  at  $830  were  sold  me  at  30,  10,  and  5  off. 
For  what  must  I  sell  them  to  gain  20  per  cent  ? 

11.  A  bill  of  goods  at  list  prices  amounted  to  12700.  The 
discounts  were  ^  and  5^.     What  was  due  on  the  bill  ? 

(^  is  often  used  for  33^^,  i  for  50^,  and  {  for  25%.) 

12.  What  is  the  cost  of  a  bill  of  cutlery  amounting  to  $272, 
if  the  discounts  are  ^,  10,  and  5,  freight  being  $2.65,  package 
$.60  ? 

13.  Find  the  net  amount  of  a  bill  for  $19.20,  subject  to 
discounts  of  16|^,  7^^,  6^^,  and  4^. 

14.  Find  the  net  amount  of  a  bill  of  $136,  discounts  being 
50^,  10^,  and  5^.  Find  a  single  discount  equivalent  to  these 
successive  discounts. 

The  net  amount  =$58.14. 

.-.  the  single  equivalent  discount  =  $136  —  $58.14  =  $77.86. 
.-.  the  rate  of  a  single  discount      =  $77.86  -f-  $136  =  .5725  =  .57ij^. 

15.  What  single  discount  is  equivalent  to  discounts  of  33^^, 
20^,  and  1^  ? 

16.  If  the  list  price  is  $37.50,  and  the  rates  of  discount  are 
20^,  12^^,  and  6^,  find  the  cost.  What  one  rate  of  discount 
is  equivalent  to  the  several  discounts  given  ? 

17.  If  the  price  of  paper  is  $1.08  a  ream  after  discounts  of 
20^  and  10^,  what  is  the  list  price  ? 

*  18.  The  cost  of  certain  goods  is  $49.63,  and  the  discounts 
are  30^,  12^  and  6^,     Find  the  list  price. 


282  SCHOOL  ARITHMETIC. 

19.  If  a  book  costs  6^  less  when  the  marked  price  is  dis- 
counted 50^  than  when  discounted  40^  and  lOfo,  what  is  the 
marked  price  ? 

20.  A  retail  dealer  buys  certain  goods  at  discounts  of  30^, 
10^,  and  8^,  and  sells  them  at  list  prices.  What  per  cent 
does  he  gain  ? 

21.  Show  that  the  discounts  15^,  10^,  and  3^  are  equiva- 
lent to  the  discounts  3^,  10^,  and  15^.  but  not  to  the  single 
discount  15^  +  10^  +  3fc. 

22.  Two  rival  houses  used  the  same  list  prices.  The  Union 
Company  offered  50  and  50  off,  while  the  National  Com- 
pany offered  50,  40,  and  10  off.  The  house  receiving  the 
higher  prices  for  its  goods  got  what  per  cent  more  than 
the  other  ? 

COMMISSIOI^. 

411.  1.  An  agent  sold  $100  worth  of  books,  and  received 
40^  of  the  sales.     How  much  did  he  get  for  his  services  ? 

2.  If  a  commission  merchant  sells  1200  worth  of  eggs  and 
charges  5^  of  the  sales,  how  much  does  he  receive  for  his 
services,  or  how  much  is  his  commission  f 

3..  How  much  should  be  received  by  an  agent  for  selling 
$500  worth  of  goods,  if  di  is  allowed  for  his  commission  ? 
How  much  will  be  realized  after  paying  the  commission,  or 
how  much  will  be  the  7iet  proceeds  9 

4.  If  an  agent  charges  5  cents  fOr  each   $1  expended  for 

butter,  what  will  a  dollar's  worth  of  butter  cost  a  grocer  who 

buys  through  the  agent  ?     If  each  dollar's  worth  costs  the 

grocer  $1.05,  how  many  dollars'  worth  can  be  bought  for 

$105? 

412.  A  person  or  firm  employed  to  transact  business  for 
another  is  called  an  Agent. 

413.  An  agent  who  sells  or  buys  produce  or  other  mer- 
chandise is  called  a  Commission  Merchant. 


PERCENTAGE.  283 

414.  The  charge  made  by  an  agent,  or  the  sum  paid  him 
for  liis  services,  is  called  his  Commission. 

Commission  is  reckoned  as  a  certain  per  cent  of  the  amount  paid  in 
buying  or  realized  in  selling.  Sometimes,  however,  it  is  a  specified  sum 
for  a  given  transaction. 

415.  The  name  Consignment  is  given  to  goods  sent  to  a 
commission  merchant  to  be  sold.  The  sender  is  called  the 
Consignor,  and  the  agent  to  whom  they  are  sent,  the  Consignee. 

416.  Tlie  sum  realized  less  the  commission  and  other 
charges  is  the  Net  Proceeds. 

In  selling  goods  consigned  to  him,  a  commission  merchant  deducts 
his  commission  from  the  sum  realized,  and  remits  the  proceeds  to  the 
consignor.  In  buying,  he  charges  the  sum  paid  plus  his  commission, 
and  both  are  included  in  the  funds  remitted  to  him  by  his  customer. 

WRITTEN    EXERCISES. 

417.  1.  A  commission  merchant  sold  a  consignment  of 
wheat  for  $7240,  charging  2^'^  commission.  What  was  his 
commission  ?     How  much  did  he  remit  to  the  consignor  ? 

2.  An  agent  receives  $510  with  which  to  purchase  goods, 
after  deducting  his  commission  of  2^.  Wha^  was  the  cost  of 
the  goods  ?     What  the  commission  ? 

$510  =  100^  +  2%,  or  102^  of  cost. 
On  what  is  the  2%  commission  reckoned  ? 

3.  A  commission  merchant  sold  140  barrels  of  flour  at 
$4.80  a  barrel.     What  was  his  commission  at  3^^  per  cent  ? 

4.  If  I  receive  25^  for  selling  500  bushels  of  wheat  at  87^ 
cents  a  bushel,  what  sum  do  I  remit  to  my  employer  ? 

5.  An  architect  charges  f^  for  plans  and  specifications, 
and  1|^  for  superintending  the  construction  of  a  building 
which  cost  $16000.     W^hat  is  his  commission  ? 

6.  A  lawyer  charged  $14  for  collecting  $200.  What  was 
his  rate  of  commission  ? 

7.  My  commission  for  selling  $792  worth  of  goods  was 
$23.76.     What  rate  did  I  charge  ? 


S84  SCHOOL  Atllf  HMETIO. 

8.  A  publisher  received  $100  as  the  proceeds  of  a  sale  of 
books.  For  how  much  were  the  books  sold,  the  agent's  com- 
mission being  20^  ? 

9.  An  agent  buys  an  invoice  of  47,500  lb.  of  coffee  at  13 
cents,  8;^  off  for  cash.     What  is  his  commission  at  2fo  ? 

10.  A  commission  merchant  remits  $427.50  as  the  net  pro- 
ceeds of  a  sale  of  100  bags  of  chestnuts,  containing  2  bushels 
each,  his  commission  being  bfo.  At  what  price  a  bushel 
did  he  sell  them  ? 

11.  A  commission  merchant  sold  produce  for  $2600,  and 
after  deducting  his  commission  and  $70  for  otl}er  charges,  he 
remitted  to  the  consignor  $2400.  What  was  his  rate  of  com- 
mission ? 

12.  A  commercial  traveler  selling  goods  at  a  commission  of 
2^^  had  an  income  in  a  certain  year  of  $2500.  What  was 
the  amount  of  his  sales  ? 

13.  My  principal  instructed  me  to  invest  $2440  in  wool, 
and  sent  me  a  draft  for  $2440  plus  2^  commission.  For 
what  amount  was  the  draft  drawn  ? 

14.  A  collector's  commission  for  collecting  taxes,  at  l^fo, 
was  $413.10.     What  sum  did  he  collect  ? 

15.  A  grain  dealer  received  $70  as  his  commission  on  the 
sale  of  a  consignment  of  wheat,  the  rate  of  commission  being 
2|^^.  If  he  sold  the  wheat  at  80^  a  bushel,  how  many 
bushels  did  he  sell  ? 

16.  A  real  estate  agent  bought  10  town  lots  at  $850  each, 
at  a  commission  of  IJ^,  and  paid  $4.25  a  lot  for  the  examina- 
tion of  title.     How  much  did  the  lots  cost  the  principal  ? 

17.  How  many  tons  of  hay  at  $14  a  ton  must  a  commission 
merchant  sell  so  that  he  may  remit  to  the  shipper  $175.63 
after  deducting  a  commission  of  3|  per  cent  ? 

18.  A  merchant  sent  goods  to  an  agent  to  be  sold  on  com- 
mission. The  latter  sold  them  and  with  the  money  bought 
1805  bbl.  of  flour  at  $5  a  barrel  after  deducting  5^  for  selling 
and  bfo  for  buying.     What  was  his  total  commission  ? 


PERCENTAGE.  285 


PROFIT   AND    LOSS. 

418.  1.  A  man  sold  a  horse  that  cost  llOO  at  a  profit  of 
25^.     How  much  did  he  gain  ?     What  was  the  selling  price  ? 

2.  How  many  dollars  are  lost  hy  selling  a  lot  that  cost  me 
$1000  at  a  loss  of  5^  ?    How  much  do  I  get  for  it  ? 

3.  A  merchant  buys  calico  for  5  cents  a  yard,  and  sells  it 
for  8  cents  a  yard.  What  part  of  the  cost  is  gained  ?  What 
per  cent  does  he  make  ? 

4.  Flour  that  cost  $5  a  barrel  was  sold  for  $4  a  barrel. 
What  was  the  loss  per  cent  ? 

6.  A  man  sold  a  saddle  for  $12  and  gained  20^,  or  ^  of  the 
cost.     What  was  the  cost  ? 

6.  A  sleigh  sold  for  124  at  a  loss  of  20^.  What  was  its 
cost? 

7.  A  dealer  sells  corn  at  a  profit  of  12^  a  bushel  and  gains 
20^.     What  did  it  cost  ? 

8.  Find  the  per  cent  gained  on  oil  bought  at  12^  and  sold 
at  14^. 

9.  What  must  be  the  selling  price  of  coffee  that  cost  20^  in 
order  to  gain  25^  ? 

10.  By  selling  tea  at  88^  a  pound,  10^  of  the  cost  was 
gained.     What  did  the  tea  cost  ? 

11.  What  per  cent  is  gained  on  goods  sold  at  double  the 
cost  ? 

12.  How  shall  I  mark  goods  that  cost  $6,  so  that  I  may 
deduct  10^  from  the  marked  price,  and  yet  make  50^  of  the 
cost  ? 

419.  Profit  is  the  excess  of  the  selling  price  over  the 
cost. 

420.  Loss  is  the  amount  by  which  the  selling  price  is  less 
than  the  cost. 

421.  The  base  upon  which  the  rate  of  gain  or  rate  of  loss 
is  estimated  is  always  the  cost  price. 


286  SCHOOL  ARITHMETIC. 

422.  The  simple  problems  of  profit  and  loss  correspond  to 
the  three  general  problems  of  percentage  : 

1.  Given,  cost  and  rate  of  gain  or  loss;  required  thegain  or  loss. 
3.  Given,  cost  and  gain  or  loss;  required  the  rate  of  gain 

or  loss. 

3.  Given,  gain  or  loss  and  rate  of  gain  or  loss ;  required 
the  cost. 

To  these  we  add  the  special  problem  : 

4.  Given,  seUi?ig  price  and  rate  of  gain  or  loss  ;  required 
the  cost. 

WRITTEN    EXERCISES. 

423.  1.  What  is  gained  by  selling  a  horse  which  cost  $120 
at  a  profit  of  33^  per  cent  ? 

2.  A  gentleman  bought  a  house  for  $1750,  and  sold  it  for 
$2050.     What  was  his  rate  of  gain  ? 

3.  If  a  merchant  sells  goods  at  a  profit  of  10^  and  gains 
$300,  what  was  the  cost  of  the  goods  ? 

4.  If  the  selling  price  of  a  horse  is  $50  and  the  loss  is  50^, 
what  is  the  cost  ? 

5.  At  what  price  must  goods  costing  $200  be  marked  in 
order  that  there  may  be  a  profit  of  20^  after  a  reduction  of 
20^  has  been  made  ? 

20^  of  $200  =  $40,  gain. 
$200  +  $40  =  $240,  selling  price. 
100^  —  20^,  or    80^  of  marked  price  =  selling  price. 
.-.80^  "  "     =$240. 

1%    .      /'  "     =     $3. 

100^  "  ''     =  $300. 

6.  At  what  per  cent  above  cost  must  goods  be  marked  in 
orderthat  a  deduction  of  20^  may  leave  20^  profit  ? 

cost  +  20^  of  cost  =  120^  of  cost,  the  selling  price. 
-  100^  —  20^,  or  80^  of  marked  price  =  selling  price. 

,' .  80^  of  marked  price  =  120^  of  cost. 
\%  "  '*     =1.5jg  of  cost. 

'  -  100^  "  "     =  150^  of  cost. 

Hence,  150^  —  100^  =  50^  above  cost. 


PERCENTAGE.  287 

7.  A  carriage  that  cost  $128  was  sold  at  a  loss  of  18fji^. 
What  was  the  amount  received  for  it  ? 

8.  Paid  $4.80  for  a  barrel  of  flour  and  sold  it  for  $6.00. 
What  per  cent  was  gained  ? 

9.  A  grocer  sells  tea  at  30  cents  a  pound  less  than  cost  and 
loses  33^^.     What  was  its  cost  ? 

10.  Paper  that  cost  $2.40  a  ream  is  sold  at  18  cents  a  quire. 
What  is  the  gain  per  cent  ? 

11.  At  what  price  must  I  sell  goods  that  cost  $f  to  gain 
20^? 

12.  What  per  cent  was  lost  on  a  wagon  which  cost  $90  and 
sold  for  $75  ? 

13.  A  boy  had  2  goats  which  he  sold  for  $6.00  each.  What 
did  they  cost  him  if  he  gained  20^  on  one  and  lost  20^  on 
the  other  ? 

14.  What  must  be  paid  for  27  bushels  of  apples  so  that 
when  sold  at  20  cents  a  half-peck  there  may  be  a  gain  of  20j]^  ? 

,  15.  A  drover  sold  40  head  of  cattle  for  $1820,  which  was 
16|^  more  than  they  cost.  What  was  the  average  cost  of 
each  ? 

16.  A  grocer  sold  potatoes  for  $2.80  a  barrel  and  made 
16f^.  If  he  had  sold  them  at  $3.20  a  barrel,  how  much  per 
cent  would  he  have  made  on  tlie  cost  price  ? 

17.  What  per  cent  is  gained  by  selling  15  ounces  of  tea  for 
a  pound  ? 

18.  If  20^  was  gained  on  flour  when  sold  at  $6  a  barrel, 
what  per  cent  was  gained  when  sold  at  $7  a  barrel  ? 

19.  If  -f  of  an  acre  of  land  was  sold  for  what  the  whole 
cost,  what  was  the  gain  per  cent  ? 

20.  When  goods  are  sold  at  f  of  their  cost,  what  per.  cent 
is  lost  ? 

21.  If  a  grocer  pays  $2  for  G  pounds  of  tea  and  sells 
4  pounds  for  $3,  what  is  his  per  cent  of  profit  ? 

22.  If  I  sell  f  of  an  acre  of  land  for  what  J  an  acre  costs 
me,  what  per  cent  do  I  lose  ? 


288  SCHOOL  ARITHMETIC. 

23.  A  farmer  sold  a  cow  for  150,  which  was  SOfo  of  the  cost 
price.     What  was  his  loss  ? 

24.  A  druggist  gained  300^  by  retailing  quinine  at  $3.00 
an  ounce.     What  did  it  cost  him  ? 

25.  If  I  of  the  selling  price  is  gained,  what  is  the  per  cent 
profit  ? 

26.  A  merchant  marks  an  article  $2.80,  but  takes  off  5^  for 
cash.     If  his  profit  is  33^,  what  was  the  cost  of  the  article  ? 

27.  If  resin  is  melted  with  3'd^fc  of  its  weight  of  tallow, 
what  per  cent  of  tallow  does  the  mixture  contain  ? 

28.  The  sum  of  10^  of  a  number  and  5^  of  half  the  re- 
mainder is  what  per  cent  of  ^  of  the  number  ? 

29.  If  I  sell  f  of  an  article  for  the  cost  of  the  whole  of  it, 
what  per  cent  gain  do  I  make  on  the  part  sold  ? 

'    30.  If  there  is  a  gain  of  120  on  tea  at  90  cents  a  pound, 
what  would  be  the  gain  per  cent  at  84  cents  a  pound  ? 
84  :  90  =  1.12^  :  (    ). 
31.  If  I  lose  10^  by  selling  goods  at  28  cents  a  yard,  for 
what  should  they  be  sold  to  gain  20^  ? 


32.  If  from  the  retail  price  of  a  book  20^  is  deducted,  and 
a  discount  of  60^  is  made  on  the  balance,  and  then  the  book 
sells  for  $1.44,  what  is  the  retail  price  ? 

33.  Sold  two  lots  at  $2400  each,  gaining  25^  on  one  and 
losing  26fo  on  the  other.     What  was  the  entire  gain  or  loss  ? 

34.  On  two  pianos  sold  for  $550  a  dealer  gained  25^.  What 
was  the  cost  of  each  if  one  cost  20^  more  than  the  other  ? 

36.  Mr.  H  bought  50  sheep  at  $5  each.  After  10  of  them 
died,  he  sold  the  remainder  so  as  to  gain  20^  on  his  invest- 
ment.    What  price  a  head  did  he  receive  ? 

36.  How  shall  a  merchant  mark  shoes  that  cost  $2.50  so 
that  he  may  fall  20^  from  the  marked  price  and  still  make 
12^? 

37.  A  speculator  sold  5000  bushels  of  July  wheat  at  75J^, 
at  a  profit  of  6^^.     What  was  the  cost  of  the  wheat  ? 


PERCENTAGE.  289 

38.  I  bought  a  watch  for  $45,  which  was  25^  less  than  its 
real  value,  and  sold  it  for  25^  more  than  its  real  value. 
What  was  my  gain  ? 

39.  A  tradesman  marks  his  goods  to  sell  at  retail  at  40^ 
above  cost,  but  sells  to  wholesale  customers  at  12^  discount 
from  the  marked  price.  What  per  cent  profit  does  he 
make  on  the  goods  sold  at  wholesale  ? 

40.  A  beef  packer  bought  a  lot  of  cattle  for  15j^  less  than 
they  cost  the  shipper,  and  made  a  profit  of  30^  on  the  trans- 
action, gaining  $1500.  What  did  the  cattle  cost  the  ship- 
per ? 

41.  A  wholesale  clothier  marked  a  lot  of  clothing  so  that 
the  price  a  suit  was  $18.  He  discounted  this  price  20^  to  a 
retailer,  who  marked  them  to  gain  25^.  Find  the  price  of  a 
suit  at  retail. 

42.  A  grocer  mixes  in  equal  quantities  teas  costing  68 
cents,  86  cents,  and  96  cents  a  pound,'  and  sells  the  mixture 
at  90  cents  a  pound.     Find  the  gain  per  cent. 

43.  What  per  cent  is  gained  by  buying  coal  by  the.  long 
ton,  and  selling  it  at  the  same  price  by  the  short  ton  ? 

44.  A  dealer  sells  72  lb.  of  sugar  for  as  much  as  87  lb.  cost 
him.     What  per  cent  does  he  gain  ? 

45.  AVhat  per  cent  above  cost  must  a  merchant  mark 
goods  so  that  after  taking  15^  from -the  marked  price  he  will 
lose  5^  ? 

46.  A  merchant  sells  goods  to  a  customer  at  a  profit  of 
60^,  but  the  buyer  becomes  bankrupt  and  pays  only  70  cents 
on  the  dollar.  What  per  cent  does  the  merchant  gain  or 
lose  on  the  sale  ? 

TAXES. 

424.  Charges  imposed  by  law  upon  persons  or  property 
for  the  support  of  government  are  called  Taxes. 
The  annual  yield  of  these  charges  is  called  revenue. 
19 


290  SCHOOL  ARITHMETIC. 


State  and  Local  Taxes. 

425.  The  charges  levied  by  a  state  for  the  feUpport  of  its 
government  are  called  State  Taxes ;  those  levied  by  the 
various  minor  civil  divisions  are  called  Local  Taxes.     The 

latter  include  county,  city,  and  township  taxes. 

426.  The  tax  on  property  is  reckoned  at  a  certain  per 
cent  of  its  assessed  value.  The  rate  of  taxation  is  usually 
expressed  as  a  certain  number  of  mills  on  the  dollar. 

427.  In  addition  to  the  tax  on  property,  some  states  levy 
an  equal  tax  upon  each  adult  citizen.  This  is  called  a  Poll 
Tax,  and  is  usually  $1  a  year. 

1.  If  taxes  are  not  paid  when  due,  the  law  usually  requires  the  de- 
linquent to  pay  a  certain  per  cent  of  his  tax  additional. 

2.  The  collector  of  taxes  usually  receives  a  per  cent  of  the  sum  col- 
lected, i.e.,  a  commission. 

WRITTEN   EXERCISES. 

428.  1.  A  tax  of  $3600  is  to  be  raised  in  a  town,  the  taxable 
property  of  which  is  valued  at  $712500,  and  there  are  750  per- 
sons subject  to  a  poll  tax  of  $1  each.  What  is  the  rate  of  taxa- 
tion ?     What  is  A's  tax,  if  he  owns  property  assessed  at  $5900. 

$3600  -  $750  =  $2850,  to  be  levied  on  property. 

The  tax  on  $712500  =  $2850. 

.-.  the  rate  of  taxation  =  $2850  h-  $712500  =  .004, or  |^. 

.-.  the  property  tax  on  $5900  =  ^%  of  $5900  =  $23.60 

$23.60  +  $1.00  =  $24.60,  A's  entire  tax. 

2.  The  assessed  value  of  a  property  is  $7500,  and  the  rate 
of  taxation  \fc.     What  is  the  tax  ? 

3.  The  assessed  valuation  of  taxable  property  in  a  town  is 
$3,500,000.  The  tax  to  be  raised  is  $4900.  What  is  the  rate 
of  taxation  ? 

4.  What  sum  must  be  assessed  to  build  a  schoolhouse  at  a 
cost  of  $5460,  and  pay  2|^  for  collecting  ? 

Compare  "buying"  problems  in  Commission. 


PERCENTAGE.  291 

6.  What  tax  will  be  paid  by  u  man  whose  house  is  valued 
at  $5160,  personal  property  at  $7815,  and  wlio  pays  a  poll 
tax  of  $1,  the  rate  of  taxation  being  J^,  or  2^  mills  on  the 
dollar  ? 

6.  At  5  mills  on  a  dollar,  how  much  is  the  tax  of  a  man 
who  owns  a  farm  of  150  acres,  worth  $50  an  acre,  but  as- 
sessed for  only  f  of  its  value  ?  If  he  does  not  pay  the  tax 
when  due  and  a  penalty  of  5^  additional  is  added,  what  is 
his  tax  ? 

7.  Find  the  tax  levy,  total  valuation,  and  poll  tax  in  the 
town,  city,  or  county  in  which  you  live,  and  compute  the 
tax  rate. 

8.  When  the  tax  rate  is  reduced  from  7  mills  to  5^  mills  on 
the  dollar,  my  taxes  are  lowered  $75.  For  how  much  am  I 
assessed  ? 

9.  The  taxes  assessed  in  a  town  are  $24000.  If  IJj^  com- 
mission is  paid  for  all  taxes  actually  collected  and  5^  of  the 
taxes  can  not  be  collected,  what  are  the  net  proceeds  ? 

10.  The  expenses  of  a  city  are  $7,500,000  per  annum,  and 
the  assessed  valuation  of  its  property  $200,000,000.  What 
must  the  tax  rate  be,  allowing  1^  of  the  taxes  to  be  uncol- 
lectible ? 

11.  I  buy  a  lot  for  $500  and  build  a  house  on  it  for  $3000. 
I  pay  a  tax  on  the  whole  of  7  mills  on  a  dollar,  the  property 
valuation  being  f  of  the  cost.  For  how  much  must  I  rent 
the  house  and  lot  to  realize  10^  a  year  on  my  money  ? 


National  Taxes. 

429.  Taxes  levied  by  the  general  government  upon  goods 
imported  from  other  countries  are  called  Duties  or  Cus- 
toms.    These  are  indirect  taxes. 

The  customs  revenue  is  collected  at  custom  houses  situated  at  ports  of 
entry  established  by  law. 


292  SCHOOL  ARITHMETIC. 

430.  When  the  duty  is  a  certain  per  cent  of  the  cost  of 
the  imported  goods  it  is  called  an  Ad  Valorem  Duty. 

431.  When  the  duty  is  a  fixed  charge  on  the  quantity  of 
goods,  without  regard  to  their  cost,  it  is  called  a  Specific 
Duty. 

Thus,  $15  a  ton  is  a  specific  duty,  while  30^  (of  cost  where  bought)  is 
ad  valorem.  . 

N"oTES.— 1.  Some  goods  are  by  law  subject  to  both  ad  valorem  and 
specific  duty. 

2.  A  schedule  of  the  legal  rates  of  duties  on  imports  is  called  a 
tariff. 

432.  The  expenses  of  the  general  government  are  met 
partly  by  customs  revenue,   partly  by  Internal  Revenue. 

The  latter  is  derived  chiefly  from  the  sale  of  licenses  to 
manufacture  or  sell  certain  domestic  articles,  as  tobacco, 
whisky,  etc. 

Note. — In  order  to  meet  war  expenditures  the  general  government 
sometimes  levies  a  special  War  Revenue  Tax,  as  in  the  recent 
Spanish-American  War  (1898).  The  revenue  is  derived  from  the  sale  of 
revenue  stamps,  which  are  required  to  be  used  on  certain  articles,  as 
checks,  mortgages,  stock  certificates,  patent  medicines,  etc. 

WRITTEN     EXERCISES. 

433.  1.  What  is  the  duty,  at  24  cents  a  pound,  on  42 
boxes  of  raisins,  each  containing  40  pounds,  and  costing  7 
cents  a  pound  ? 

Note. — The  rates  of  duty  given  here  are  from  the  Dingley  Tariff 
Law,  which  went  into  effect  July  24,  1897. 

2.  What  will  be  the  duty,  at  25^  ad  valorem,  on  a  shipment 
of  books  from  London  to  New  York,  invoiced  at  $4500  ? 

3.  What  is  the  duty  on  4368  pounds  of  wool  invoiced  at 
$1826,  when  the  rate  is  11  cents  a  pound  ? 

4.  What  is  the  duty  on  a  ton  of  tin  plate  at  1^  ct.  a 
pound  ? 


REVIEW   WORK.  293 

5.  What  is  the  duty  on  300  lb.  of  perfumery  that  cost  $2 
a  pound,  when  the  duty  is  $.60  a  pound  and  45^  ad  valorem  ? 

6.  If  200  yards  of  silk  are  purchased  at  $1 .25  a  yard,  what  is 
the  duty  at  60^  ad  valorem?    What  specific  duty  does  it  equal  ? 

7.  Imported  100  half-pint  bottles  of  champagne.  If  the 
duty  is  12  a  dozen,  breakage  5;?^,  what  is  the  duty  ? 

8.  I  paid  40^  duty  on  a  watch  which  cost  me,  including 
the  duty,  160.  If  it  had  been  on  the  free  list  how  much  less 
would  it  have  cost  me  ? 

9.  A  manufacturer  produces  oleomargarine  at  a  cost  of  3 
cents  a  pound.  What  per  cent  does  he  gain  if  he  sells  it  at 
20  ct.  a  pound  after  paying  tlie  internal  revenue  tax  of  15 
cents  a  pound  ? 

10.  A  dealer  imported  20000  oriental  rugs,  each  2  ft.  6  in. 
by  6  ft,,  and  invoiced  at  $1.80.  The  duty  was  10  cents  a 
square  foot  and  40^  ad  valorem.  How  much  did  the  govern- 
ment get  as  a  result  of  this  importation  ? 

REVIEW   WORK. 

ORAL    EXERCISES. 

434.  1.  Mr.  B  bought  a  cow  for  $30,  which  was  50^  of 
his  money.     How  much  had  he  left  ? 

2.  A  man  gave  25^  of  his  money  for  an  organ  valued  at 
$125.    How  much  money  had  he  left  ? 

3.  A  laborer  who  earns  $1.50  a  day  saves  20^  of  it.  How 
much  does  he  spend  ? 

4.  A  merchant  bought  a  stove  for  $15  and  sold  it  for  $20. 
What  was  the  gain  per  cent  ? 

6.  If  a  grocer  gains  10^  on  eggs  that  cost  15  cents  a  dozen, 
what  price  is  paid  by  the  purchaser  ? 

6.  When  a  shop-worn  article  that  cost  $2.80  is  sold  for  25^ 
les^  than  cost,  what  is  received  for  it  ? 

7.  What  per  cent  do  I  gain  by  selling  for  $15  a  cart  tb^t 
cost  $12  ? 


294  SCHOOL  ARITHMETIC. 

8.  Thirty  dollars  is  37^^  of  what  I  paid  for  a  wagon.  How 
much  less  than  $100  did  I  pay  ? 

9.  What  per  cent  of  a  yard  is  2  feet  ? 

10.  In  a  certain  district  there  are  20  school  days  in  a 
month.  When  a  boy  is  absent  5  days  each  month,  what  per 
cent  of  the  time  does  he  lose  ? 

11.  When  pencils  that  cost  1  cent  each  are  sold  for  3  cents 
each,  what  is  the  gain  per  cent  ? 

12.  I  bought  a  lot  for  1250,  and  through  an  agent  sold  it 
for  $300.  If  the  agent's  commission  was  5^,  how  much  did 
I  g^iii  ? 

13.  If  4  lemons  are  sold  for  what  5  cost,  what  j^er  cent  is 
gained  ? 

14.  Mr.  H  bought  wheat  at  $.80  a  bushel.  If  he  sells  it 
at  5fo  gain,  how  many  bushels  must  he  sell  to  gain  $5  ? 

15.  What  is  the  gain  per  cent  when  an  article  that  cost 
half  a  cent  is  sold  for  a  cent  and  a  half  ? 

16.  I  sold  two  sheep  for  $12  each.  On  one  I  gained  20^, 
and  on  the  other  I  lost  20^.  How  much  did  my  loss  exceed 
my  gain  ? 

WRITTEN     EXERCISES. 

435.  1.  A  man  buys  a  quantity  of  wine  at  $5  a  gallon  ; 
20^  of  it  is  wasted.  At  what  price  a  gallon  must  he  sell  the 
remainder  so  as  to  gain  20^  on  his  outlay  ? 

2.  Sold  72  yards  of  carpet  at  $1.37^  a  yard,  and  thereby 
gained  $18.     What  per  cent  did  I  gain  ? 

3.  The  difference  between  28^  and  59^  of  a  number  is325.5. 
What  is  the  number  ? 

4.  Fifteen  per  cent  of  484  is  66^  of  what  number  ? 

6.  How  much  land  at  $35  an  acre  can  an  agent  buy  with 
$3965.50,  after  deducting  his  commission  of  3  per  cent  ? 

6.  AVhat  must  I  ask  for  a  house  that  cost  $4000  in  order 
that  I  may  gain  12^^  on  my  purchase  after  decreasing  the 
amount  originally  asked  by  10  per  cent  ? 


REVIEW  WORK.  295 

7.  The  taxes  assessed  in  a  town  are  124000.  If  1.75j^  com- 
mission is  paid  on  all  taxes  actually  collected,  and  5^  of  the 
taxes  can  not  be  collected,  what  are  the  net  proceeds  ? 

8.  A  boy  spent  50^  of  his  money  at  one  time,  and  GOj^  of 
the  remainder  at  another,  and  had  6  cents  left.  How  much 
had  he  at  first  ? 

9.  A  watch  that  cost  $62.50  was  sold  for  $37.50.  What  per 
cent  was  lost  ? 

10.  A  farm  was  bought  for  $3090,  which  was  12^^  more  than 
its  value.     What  was  its  value  ? 

11.  Out  of  a  circle  38  inches  in  diameter  there  is  cut  a 
circle  27  inches  in  diameter.  What  per  cent  of  the  larger 
circle  is  left  ? 

12.  A  man  invested  37^^  of  his  money  in  land  at  $40  an 
acre.     If  he  had  $4500  left,  how  many  acres  did  he  buy  ? 

13.  A  bookseller  for  a  number  of  boolis  receives  $33.15, 
after  giving  a  discount  of  15^  from  the  list  prices.  What  is 
his  gain,  if  he  himself  gets  a  discount  of  25j^  ? 

14.  What  is  the  difference  on  a  bill  of  $3500  between  a  dis- 
count of  40^  and  discounts  of  30;^  and  10^  ? 

15.  A  store  was  sold  at  auction  for  $3375,  which  was  only 
y\  of  its  vahie.     What  was  the  loss  per  cent  ? 

16.  The  revenue  of  the  post-office  department  for  a  certain 
year  (1899)  was  $95,021,384,  and  the  expenditures  were 
$101,632,160.  The  excess  of  expenditures  was  what  per  cent 
of  the  revenue  ? 

17.  A^s  money  is  60;^  of  B's.  What  per  cent  of  A's  money 
is  B's? 


18.  A  section  of  land  was  bought  at  $24  an  acre. 
What  was  received  for  240  acres  when  sold  at  a  gain  of 
30^  ? 

19.  A  hardware  merchant  bought  three  dozen  agate  basins 
at  the  rate  of  3  for  $5,  and  sold  them  at  a  gain  of  $10  on  the 
whole,     What  was  the  gain  per  cent  ? 


296  SCHOOL  ARITHMETIC. 

20.  If  I  sell  f  of  an  article  for  what  f  of  it  cost,  what  per 
cent  do  I  gain  ? 

21.  How  many  pounds  of  flour  are  required  to  make  100 
2-pound  loaves  of  bread,  if  the  bread  weighs  30^  more  than 
the  flour  used  ? 

22.  What  per  cent  do  I  lose  by  selling  |  of  my  sheep  for 
what  f  of  them  cost  ? 

23.  A  coin  is  22  parts  copper  and  3  parts  nickel.  What 
per  cent  of  the  coin  is  copper  ?  What  per  cent  of  the  nickel 
is  the  copper  ? 

24.  A  man  sold  flour  for  110,  thereby  losing  20^.  How 
much  would  he  have  lost  by  selling  it  for  18  ? 

25.  Eight  pounds  of  cofPee  and  1  pound  of  tea  are  sold  at 
an  average  price  of  30  ct.  a  pound,  being  an  advance  of  35^ 
on  the  cost.  If  the  pound  of  tea  cost  $1,  what  did  the  coffee 
cost  a  pound  ? 

26.  Bought  land  at  160  an  acre.  How  much  must  I  ask 
an  acre  that  I  may  deduct  25^  from  my  asking  price,  and  yet 
make  20^  of  the  purchase  price  ? 

27.  A  man  bought  goods  at  25^  below  the  list  price,  and 
sold  them  at  20^  above  the  list  price.  How  much  did  he  in- 
vest if  he  gained  $270  ? 

28.  An  agent  received  $5250  to  invest  at  5^  commission. 
If  he  reserves  his  commission,  and  invests  the  remainder  in 
wheat  at  $.75  a  bushel,  how  many  bushels  will  be  purchased  ? 

SUPPLEMENTARY     EXERCISES     (FOR     ADVANCED     CLASSES). 

436.  1.  What  per  cent  of  a  bill  does  a  merchant  receive  if 
he  gives  a  discount  of  20^,  10^,  and  5fo  ? 

2.  A  man  bought  a  carriage  for  $225.  One  fifth  of  the 
cost  of  the  carriage  was  9fo  of  what  he  paid  for  a  span  of 
horses..    Find  the  cost  of  both. 

3.  A  merchant  pays  $1800  a  year  rent  for  a  storeroom  ; 
45^  of  this  sum  is  18^  of  one  half  of  his  profit,  What  is  his 
profit  ?     . 


REVIEW   WORK.  297 

4.  Wliat  is  the  rate  of  gain  when  25^  of  the  selling  price  is 
gain  ? 

5.  Mr.  A  bought  a  watch  for  $30,  which  was  40^  less  than 
its  value,  and  sold  it  for  50^  more  than  its  value.  How  much 
did  he  gain  ? 

6.  Josie  paid  $100  for  her  pony,  which  was  15^  more  than 
double  the  amount  she  paid  for  her  cart.  What  did  she  pay 
for  both  ? 

7.  An  article  that  cost  nothing  was  sold  for  $5.20.  What 
was  the  gain  per  cent  ? 

8.  Mr.  G  said  he  bought  a  book  for  $2  and  sold  it  at  a  loss 
of  100^.     Did  he  state  two  facts  or  but  one  ?     Why  ? 

9.  A  farmer  sold  a  horse,  [it  such  a  price  that  f  of  the  gain 
equaled  |  of  the  cost.     What  was  his  gain  per  cent  ? 

10.  A  nierchant's  private  key  for  marking  goods  is  "  p  r  e  - 
caution."  How  must  he  mark  cloth  that  cost  10  cents 
a  yard  so  as  to  gain  70,^  ? 

11.  How  much  water  must  be  mixed  with  31  gallons  1 
quart  of  alcohol,  which  cost  $45  25,  so  that  the  mixture  may 
be  sold  at  $1.25  a  gallon,  and  25^  gained  ? 

12.  A's  money  is  12^  of  B's  and  16fc  of  C's.  B  has  $100 
more  than  C.     How  much  has  A  ? 

13.  The  United  States  produced  91,070  metric  tons  of 
zinc  in  a  certain  year  (1897),  and  that  produced  by  other 
countries  constituted  79.56^  of  the  total  production.  What 
was  the  total  production  ? 

14.  If  15^  can  be  saved  by  employing  women  when  men 
are  paid  $1.60  a  day,  and  if  a  man  does  -J  more  than  a  woman 
in  the  same  time,  what  are  the  daily  wages  paid  to  women  ? 

15.  If  water  expands  7^^  in  freezing,  what  will  a  cubical 
block  of  ice  weigh  that  measures  3  dm.  on  an  edge  ? 

16.  The  operatives  in  a  certain  factory  have  their  working 
hours  reduced  from  10  hours  to  8  hours  a  day  without  any 
reduction  in  their  daily  wages.  By  what  per  cent  are  their 
daily  wages  increased  by  this  change  in  time  ? 


INTEREST. 

437.  1.  I  borrow  $100  and  agree  to  pay  $6  for  the  use  of 
it  for  one  year.  What  should  I  pay  if  I  use  the  money  2 
years  ? 

2.  I  can  borrow  money  by  paying  5  cents  for  the  nse  of  $1 
for  a  year.  What  must  I  pay  for  the  use  of  $100  at  this 
rate  ? 

3.  If  I  pay  6  cents  for  the  use  of  $1  for  1  year,  what  should 
I  pay  for  its  use  for  3  years  ?  For  85  for  1  year  ?  For  $5 
for  two  years  ?     For  $100  for  a  year  ? 

4.  I  borrow  $100  and  agree  to  pay  back  the  money  at  the 
end  of  a  year,  and  G^  additional  for  its  use.     How  much  do 

1  pay  for  the  use  of  the  money,  or  how  much  interest  do  I 
pay  ?     How  much  do  I  pay  in  all  at  the  end  of  the  year  ? 

5.  What  should  I  receive  for  a  loan  of  $500  for  2  years 
at  6^  ?  What  interest  should  I  receive  for  a  loan  of  $300  for 
5  years  at  the  same  rate  ? 

438.  Interest  is  a  sum  paid  for  the  tise  of  money. 

439.  The  money  for  the  use  of  which  interest  is  paid  is 
called  the  Principal. 

440.  The  sum  of  the  principal  and  interest  is  the  Amount. 

441.  The  Rate  of  Interest  is  always  expressed  as  a  cer- 
tain per  cent  of  the  principal.     The  unit  of  time  is  the  year. 

Thus,  interest  at  6%  means  that  the  charge  for  a  year  is  6%  of  the 
principal.     Hence  for  6  months  the  charge  is  S%  ;  for  2  months,  1%  ;  for 

2  years,  12^. 

Notes. — 1.  In  ordinary  interest  it  is  customary  to  regard  a  year  as  13 
moi^ths,  and  a  month  as  30  days. 


INTEREST.  299 

2.  The  mills  in  the  interest  or  auiount  are  usually  disregarded  unless 
they  are  5  or  more,  in  which  case  they  are  regarded  as  a  cent. 

442.  lip  stands  for  the  principal,  r  for  the  rate  of  inter- 
est, t  for  the  number  of  years,  and  i  for  tlie  interest,  we  have 

the  equation — 

i  =:p  X  r  X  t. 

Queries. — Which  is  the  product  f    Which  mr^  factors  f 

WRITTEN    EXERCISES. 

443.  1.  What  is  the  interest  on  $250  for  3  years  at  5j^  ? 

Interest  for  1  year  =  ti%  of  |250  =  $12.50. 
Interest  for  3  years  =  3  x  $12.50  =  $37.50. 
Have  the  pupil  solve  by  substituting  in  the  equation  given  above. 

2.  What  is  the  interest  on  $350  for  3  years  8  months  at  6j^  ? 
(3  yr.  8  mo.  =  3|  yr.) 

Find  the  interest  on  : 

3.  $480  for  3  years  at  6^. 

4.  $531.21  for  5  years  at  8^. 

5.  $375.75  for  8  years  at  5^^. 

6.  $870.30  for  3  years  at  6^. 

7.  $427.30  for  1  year,  3  months  at  5^. 

8.  $150.25  for  6  years,  1  month  at  10^. 

9.  $325.75  for  3  years,  11  months  at  4^^. 

10.  $650.00  for  7  years,  7  months  at  6^. 

11.  $860.00  for  6  years,  3  months  at  4,^. 

12.  $476.38  for  4  years,  8  months  at  6^. 

13.  $410.30  for  7  years,  3  months  at  6^. 

14.  $367.50  for  8  years,  7  months  at  5^. 

15.  $1.50  for  5  years,  4  months  at  8^. 

16.  Find  the  interest  on  $150  for  3  months  at  6^. 

Interest  for  one  year        =  Q%  of  the  principal. 

"         "    two  months  =  1%  of  principal  =  $1.50. . 
"    one  month    =  i  of  $1.50  =       .75. 


Interest  for  3  months      ==  $2.2$, 


300  SCHOOL  ARITHMETIC. 

17.  What  is  the  interest  on  1180  for  98  days  at  6^  ? 

Interest  for  60  days  =  1%  of  principal  =  $1.80. 

"      "    30     **     =  i  of  $1.80  =       .90. 

"       "      6     '*     =:iof90^  =      .18. 

"      "  _2     ''     =i  of  18^  =      .06. 

Interest  for  98  days  =  $2.94. 

18.  Find  the  interest  on  $860  for  7  months,  24  days  at  5fj. 

Interest  for  60  da.  (2  mo.  )  =  1^  of  $860  =  $8.60. 
"  120  da.  (4  ino.  )  =  2  x  $8.60  =  17.20. 
"  30da.  (1  rao.  )  =  i  of  $8.60=  4.30. 
"      24  da.  (.8mo.)  =  .8  of  $4.30  =    3.44. 

Interest  for  7  mo.  24  da.  at  Q%  =  $33.54. 

The  interest  at  1%  =  $5.59. 

The  interest  at  5%  =  $27.95. 

It  may  often  be  more  convenient  to  proceed  as  follows  :  1 

Interest  for  2     mo.  =  1%  of  $860   =    $8.60. 
"  1     mo.  =  i     of  $8.60  =    $4.30. 
"  7.8  mo.  =  7.8  x  $4.30  =  $33.54. 
Interest  at  5%  found  as  above. 
Let  it  be  observed  that  the  interest  at  any  rate  may  be  found  from  the 
interest  at  6%. 

Find  the  interest  at  6^  on  : 

19.  $250  for  6  months.  31.  $840  for  5  months,  12  days. 

20.  $720  for  10  months.  32.  $7G0  for  9  months,  24  days. 

21.  $150  for  15  months.  33.  $500  for  7  months,  18  days. 

22.  $75  for  23  months.  31  $950  for  3  months,  25  days. 

23.  $240  for  G9  days.  35.  $375  for  3  years,  7  months. 

24.  $225  for  76  days.  36.  $624  for  2  yea*rs,  11  months. 

25.  $412  for  93  days.  37.  $480  for  1  yr.  4  mo.  20  da. 

26.  $1060  for  119  days.  38.  $320  for  2  yr.  7  mo.  13  da. 

27.  $87.50  for  126  days.  39.  $1  for  1  yr.  8  mo.  27  da. 

28.  $1  for  93  days.  40.  $5  for  5  yr.  5  mo.  5  da. 

29.  $272  for  29  days.  41.  $275  for  8  yr.  1  mo.  27  da. 

30.  $27.60  for  3  yr.  10  mo.     42.  $360.25  for  July,  1901. 

43.  Find  the  interest  on  $1400  from  June  6,  1896,  to  July 
6^  1900,  at  efc.  ■ 


INTEREST.  301 

44.  Find  the  amount  of  $950  from  Nov.  6,  1896,  to  June 
15,  1898,  ut  l</o. 

45.  Find  the  amount  of  $343.50  from  Jan.  24,  1895,  to 
Dec.  24,  1899,  at  5^. 

Find  the  interest  and  amount  of  : 

46.  $750  for  48  days  at  6^. 

47.  $1000  for  63  days  at  6^. 

48.  $000  for  225  days  at  5^. 

49.  $120  for  93  days  at  7^. 

50.  $360  for  123  days  at  3^^. 

51.  $630  for  63  days  at  4|^. 

52.  $860  for  33  days  at  8^. 

53.  $9430  for  2  yr.  5  mo.  7  da.  at  m. 

54.  $3875  for  100  days  at  4^. 

55.  $720  for  6  mo.  3  days  at  3^. 

56.  $842  from  Jan.  1  to  April  10  at  6^. 

57.  Mr.  Stoney  borrowed  $1250  for  94  days  at  6^.  How 
much  interest  did  he  have  to  pay  ? 

58.  Mr.  Thomas  bought  a  piano  for  $350,  agreeing  to  pay 
interest  at  5^  if  he  did  not  pay  for  the  instrument  in  30  days. 
He  paid  the  bill  at  the  end  of  66days,-  Whatamountdid  liepay  ? 

59.  A  man  borrowed  $6500  with  which  lie  engaged  in  a 
business  that  paid  him  a  profit  of  25;^  on  his  capital.  If  the 
loan  cost  him  6^,  what  was  his  annual  net  profit  ? 

60.  A  man  bought  a  farm  for  $4200,  agreeing  to  pay  $1000 
at  the  end  of  each  year,  and  also  to  pay  6^  interest  on  all 
unpaid  money.  At  the  end  of  four  years,  how  much  did  he 
still  owe  ? 

OLD  6  PER  CENT  METHOD. 

61.  Find  the  interest  on  $360  for  1  yr.  1  mo.  1  da.  at  6^. 
Interest  at  %%  for  1  yr.  =  .06      of  the  principal. 
Interest  at  6^  for  1  mo.                =  .005    of  the  principal. 
Interest  at  6^  for  1  da.                =  .000^-  of  the  principal. 
Interest  for  1  yr.  1  rao.  1  da.      =  .065^  of  the  principal. 

.'.  the  interest  =  .065i  x  $360  =  $33.46. 


302  SCHOOL  ARITHMETIC. 

62.  Find  the  interest  on  $400  for  8  mo.  21  da.  at  6^. 

Interest  for    8  mo.  =    8  x  .005    —  .040 

Interest  for  21  da.  =  21  x  .000^  =  .0035 

Interest  for  8  mo.  21  da.  =  .0435 

.-.  tlie  interest  =  .0435  x  $400  =  $17.40. 

E^^  Should  this  method  be  preferred  by  any  one,  it  may  be  employed 
in  computing  the  interest  in  all  or  part  of  the  preceding  examples. 

Exact  Interest. 

444.  In  the  preceding  exercises  360  days  have  been  re- 
garded as  a  year.  To  find  exact  interest,  however,  we  must 
reckon  365  days  to  a  year. 

Exact  interest  is  used  by  the  national  and  some  state  governments  in 
their  computations  of  interest,  and  also  by  some  trust  companies. 

Find  the  exact  interest  on  : 

1.  $1000  for  129  days  at  4^. 

Interest  for  a  year     =  A%  of  $1000      =  $40. 
Exact  interest  for    1  day    =  $40.00  ^  365  =  $.10958. 
"  129  days  =  129  x  $.10958  =.-  $14.14. 

2.  $450  for  228  days  at  6^.         5.  $200  for  63  days  at  5^. 

3.  $2568  for  93  days  at  M.         6.  $875  for  151  days  at  4^^. 

4.  $84.75  for  37  days  at  4^.       7.  $375  for  33  days  at  6^^ 

8.  $725  from  April  1  to  July  19  at  6^. 

PROMISSORY    NOTES. 

445.  A  promise  in  writing  to  pay  a  stated  sum  of  money 
on  demand  or  at  a  specified  time  is  called  a  Promissory 
Note. 

1.  The  person  who  signs  the  note  is  called  the  maker. 

2.  The  person  to  whom  the  note  is  made  payable  is  called  t\\Q  payee, 

3.  The  person  who  owns  the  note  is  called  the  holder. 


INTEREST.  303 

Forms  of  Notes. 
446. 

(a) 
$350.50.  Richmond,  Va.,  May  1,  1900. 

On  demand  I  promise  to  pay  Chas.  S.  McNulty  three  hun- 
dred fifty  and   ^^  dollars,  with   interest  at  6^,   for   value 

received. 

I.  L.  Beverage. 

(b) 
1500.  New  York,  Dec.  5,  1901. 

Thirty  days  after  date,  for  value  received,  I  promise  to  pay 
to  Henry  Wilson,  or  order,  five  hundred  dollars,  with  in- 
terest at  6^. 

John  R.  Logan. 

(c) 
1280.  Chester,  Pa.,  July  31,  1899. 

Three  months  after  date,  I  promise  to  pay  G.  B.  M.  Zerr, 

or  bearer,  two  htindred  eighty  dollars,  value  received. 

M.  G.  Brumbaugh. 

1.  A  note  payable  on  demand  is  called  a  demand  note. 

2.  A  note  payable  at  a  specified  time  is  called  a  time  note,  and,  if 
the  words  "with  interest"  are  omitted,  does  not  bear  interest  until 
it  is  clue. 

3.  The  sum  of  money  named  in  a  note  is  called  the  face.  However, 
in  the  case  of  notes  with  interest  the  practice  is  not  uniform,  many  lead- 
ing bankers  calling  the  sum  due  at  maturity  the  face. 

447.  A  note  made  payable  to  the  order  of  the  payee  or  to 
the  bearer  is  called  a  Negotiable  Note ;  that  is,  one  that  can 
be  bought  and  sold.  A  note  made  payable  to  the  payee  only 
is  called  a  Non-negotiable  Note  ;  that  is,  one  that  can  not 
be  bought  or  sold. 

1.  Which  of  the  notes  given  above  are  negotiable  ? 

2.  What  words  must  a  note  contain  in  order  to  be  nego- 
tiable ? 


804 


SCHOOL  ARITHMETIC. 


448.  A  person  who  places  his  signature  on  the  back  of  a 
negotiable  note  is  called  an  Indorser. 

1.  A  note  that  is  payable  to  the  person  named  therein  "or  bearer" 
may  be  sold  by  the  payee  and  transferred  by  delivery. 

2.  A  note  payable  to  a  specified  person  "  or  order  "  may  also  be  sold 
by  the  payee,  but  the  transfer  is  indicated  by  his  indorsement. 

3.  A  special  indorsement  specifies  the  person  to  whose  order  the  note 
is  to  be  payable,  and  the  indorsement  of  such  person  is  in  turn  necessary 
to  the  further  sale  and  transfer  of  the  note.  An  indorsement  in  blank 
specifies  no  person  to  whom  it  is  payable — the  payee  merely  writing  his 
name  across  the  back — and  a  note  so  indorsed  is  payable  to  hearer. 

4.  By  indorsing  a  note  the  payee  becomes  liable  for  its  payment,  unless 
^e  is  permitted  by  ithe  buyer  to  q^ualify  his  liability  by  adding  to  his 
signature  the  words  "  ivithout  recQurse.'' 

449.  Where  days  of  grace  are  not  allowed,  a  note  is 
payable  at  the  time  specified  therein  ;  but  about  one  half  of 
the  states  allow  three  ^'  days  of  grace  ''  for  the  payment  of  ne- 
gotiable notes  before  they  are  said  to  mature,  or  be  legally  due. 

Thus,  a  note  nominally  due  Sept.  2  is  legally  due  Sept.  5,  and  matu- 
rity is  indicated  thus  :  "  Due  Sept.  %;," 

1.  In  ascertaining  the  date  of  maturity  when  the  time  is  given  in  days, 
we  count  forward  the  exact  number  of  days  from  the  date  of  the  note  ; 
when  the  time  is  given  in  months,  we  count  forward  by  calendar  months. 

2.  When  the  time  is  given  in  months,  a  note  dated  on  the  last  day  of 
a  "  long  "  month  falls  due  on  the  last  day  of  the  proper  month.  Thus, 
a  note  dated  Jan.  31,  1900,  is  due  Feb.  28  if  .for  one  month,  and  April 
30  if  for  three  months. 

3.  For  further  information  as  to  the  date  of  maturity,  see  Bank 
Discount. 

4.  Grace  has  been  abolished  in  the  following  : 


California, 

Maine,* 

Wisconsin, 

Colorado, 

Maryland, 

Ohio, 

Connecticut, 

Massachusetts,* 

Oregon, 

Dist.  of  Columbia, 

Montana, 

Pennsylvania, 

Florida, 

New  York, 

Utah, 

Idaho, 

New  Jersey, 

Vermont, 

Illinois, 

New  Hampshire,* 
North  Dakota, 
*  Except  on  Sight  Drafts. 

Virginia. 

INTEREST.  305 


WRITTEN     EXERCISES. 

450.  1.  Write  a  negotiable  note  for  $350,  with  interest  at  6^, 
payable  to  George  H.  Hugus,  making  J.  H.  Lafferty  the  maker. 

2.  Write  a  non-negotiable  note  for  $50,  making  W.  P. 
Campbell  the  payee,  and  0.  A.  Stephenson  the  maker, 
payable  on  demand. 

3.  Write  a  negotiable  note  for  $275,  payable  on  demand,  to 
Geo.  0.  Rodgers,  or  order,  with  interest  at  6^ ;  Walter  New- 
man, maker. 

4.  Write  a  negotiable  note  for  $1724,  payable  one  day  after 
date,  to  J.  C  .  Matheny,  or  bearer,  with  interest  at  G^  ;  E.  A. 
Dudley,  maker.     Indorse  in  blank. 

5.  Write  a  negotiable  note  for  $1000,  due  three  months 
after  date,  payable  to  William  Gibson,  or  order,  interest  at 
6^ ;  F.  G.  Mauzy,  maker. 

6.  Write  a  note  that  will  be  negotiable  without  indorsement, 
binding  yourself  to  pay  to  C.  L.  Magee  $125  in  two  years, 
with  interest  at  6^. 

7.  Write  a  note  that  is  negotiable  when  indorsed,  binding 
J.  C.  Kendall  to  pay  to  your  order  $275.87  on  demand,  with 
interest  at  5^. 

8.  Indorse  the  last  note  so  that  II.  II.  Dinsmore  may  sell  it  to 
James  M.  Laird.    Also  put  Mr.  Dinsmore's  indorsement  on  it. 

9.  From  the  following  data  write  a  note  bearing  interest 
from  date  :  Date,  May  24,  1900  ;  payee,  Nannie  Mackrell  ; 
amount  named,  $150  ;  rate,  6fo  ;  maker,  C.  Lamb  ;  maturity, 
August  24,  1900. 

10.  Find  the  amount  due  on  the  last  note  at  maturity. 

11.  Find  the  date  of  maturity  and  the  interest  of  the  fol- 
lowing note  : 

$250.  Boston,  March  16,  1900. 

Sixty  days  after  date  I  promise  to  pay  W.  M.  McCullough, 
or  order,  two  hundred  fifty  dollars,  with  interest  at  6^,  at  the 
Diamond  National  Bank.  Joseph  Tuener. 

30 


306  SCHOOL  ARITHMETIC. 

12.  The  following  note  was  paid  Dec.  14,  1899.  Find  the 
amount  paid  and  the  date  it  became  legally  due. 

|325.  MoNTEEEY,  Va.,  June  1,  1899. 

Ninety  days  after  date  I  promise  to  pay  E.  M.  Arbogast, 
or  order,  three  hundred  twenty-five  dollars,  with  interest  at 
6^,  for  value  received. 

J.  E.  Spiegel. 

13.  Write  a  negotiable  note  for  $350,  payable  to  Chas.  M. 
Loomis,  due  in  30  days  from  date,  with  interest,  and  signed 
by  0.  A.  Bird.  Indorse  properly  for  transferring  to  W.  B. 
Smith,  or  order. 

PARTIAI.    PAYMENTS. 

451.  A  Partial  Payment  is  a  payment  of  a  part  of  a  note 
or  other  obligation  bearing  simple  interest. 

The  amount  and  date  of  a  payment  are  usually  written,  or  indorsed, 
upon  the  back  of  the  note. 

452.  The  method  usually  employed  in  computing  interest 
on  notes  when  partial  payments  have  been  made  is  expressed 
in  what  is  known  as  the  United  States  Rule.  It  is  the  legal 
method  in  most  states. 

453.  United  States  Eule. — 1.  From  the  amount  of  the 
principal,  computed  to  the  time  when  the  payment  or  the  sum 
of  the  payments  equals  or  exceeds  the  interest  due,  subtract 
such  payment  or  the  sum  of  the  payments. 

2.  Treat  the  remainder  as  a  new  principal,  and  proceed  as 
before. 

WRITTEN    EXERCISES. 

454.  1.  A  note  for  $600,  bearing  interest  at  6^,  and  dated 
July  10,  1897,  has  the  following  payments  indorsed  upon  it : 

Jan.  25,  1900,  $5 ;  May  20,  1900,  $100 ;  Oct.  2,  1901,  $200. 
What  is  due  March  20,  1902  ? 


INTEREST.  307 

Principal 1600.00 

Interest  to  May  20,  1900  (34i  mo.) 103.00 

Amount $703.00 

First  payment  +  second  payment 105.00 

Balance  due $598.00 

Interest  to  October  2,  1901 49.04 

Amount $647.04 

Third  payment 200.00 

Balance  due $447.04 

Interest  to  March  20,  1902 12.52 

Balance  due  March  20,  1902 $459. 56 

The  interest  from  July  10,  1897  to  Jan.  25,  1900  is  $91.50,  which  is 
more  than  the  payment  ;  hence,  by  the  rule,  it  cannot  be  added  to  the 
principal.  The  reason  for  this  is  that  if  the  interest  were  now  added  and 
the  $5  deducted,  the  new  principal  would  be  $686.50,  and  the  borrower 
would  be  paying  interest  on  $86.50  too  much.  We  therefore  compute 
interest  to  the  time  of  the  second  payment,  which  together  with  the  first 
payment  is  more  than  suflBcient  to  pay  accrued  interest. 

2.  On  a  note  for  $2000,  interest  at  6^  dated  Dec.  10,  1898, 
and  payable  in  12  mo.,  are  found  the  following  indorse- 
ments : 

Jan.  27,  1898,  $49  ;  Feb.  5,  1899,  $104  ;  May  16,  1899, 
$60  ;  July  21,  1899,  $700. 

The  note  was  paid  six  months  after  it  became  due.  What 
was  then  paid  ? 

3.  A  note  for  $698,  dated  Jan.  24,  1899,  is  indorsed  as 
follows  : 

Feb.  17,  1899,  $115  ;  Aug.  5,  1899,  $82;  Aug.  18,  1899, 
$129  ;  Oct.  11,  1899,  $213. 

At  6^,  what  is  due  Nov.  5,  1899  ? 

4.  A  note  for  $600,  dated  Aug.  10,  1897,  and  drawing 
interest  at  5^,  has  indorsements  as  follows  : 

Feb.  4,  1898,  $50 ;  July  .27,  1898,  $10 ;  Oct.  9,  1898,  $75. 
How  much  was  due  Dec.  15,  1898  ? 


308  SCHOOL  ARITHMETIC. 

5. 

$3150.  Buffalo,  N.  Y.,  Nov.  1,  1898. 

Thirty  days  after  date,  for  value  received,  I  promise  to  pay 
T.  B.  DeArmit  three  thousand  one  hundred  fifty  dollars, 
with  interest  at  4^.  C.  R.  McDaniel. 

Indorsements  : 

Dec.  27,  1898,  $1080  ;  May  15,  1899,  $540  ;  June  21,  1899, 
$310 ;  Sept.  22,  1899,  $770. 

How  much  was  due  Oct.  21,  1899  ? 

455.  When  partial  payments  are  made  on  notes  running 
one  year  or  less,  the  balance  due  at  settlement  is  sometimes 
computed  by  what  is  known  as  the  Mercantile  Rule. 

456.  Mercantile  Rule. — From  the  amount  of  the  princi- 
pal at  the  date  of  settlement  suhtract  the  amount  of  each  payment 
at  the  same  time,  and  the  remainder  will  he  the  balance  due. 

1.  A  note  for  $250.00,  bearing  interest  at  6^,  and  dated 
July  17,  1899,  has  the  following  indorsements  : 

Sept.  20, 1899,  $80;  Jan.  1, 1900,  $50  ;  March  13,  1900,  $50. 

Find  the  balance  due  at  settlement.  May  5,  1900. 

Amount  of  principal  to  May  5,  1900 $262.63 

Amount  of  1st  payment  to  May  5,  1900 . . .  $83 .  00 
Amount  of  2nd  payment  to       **         *'    ...    51.03 
Amount  of  3d  payment  to       "          "   ...    50.43      $184.46 
Balance  due  May  5,  1900 $78.17 

2.  A  note  for  $1200,  dated  April  22,  1900,  and  due  5 
months  after  date,  bears  interest  at  8^.  The  following  pay- 
ments have  been  made  : 

May  3,  1900,  $125  ;  August  7,  1900,  $25. 
Find  the  balance  due  at  maturity. 

3.  A  note  for  $850  was  made  Jan.  18,  1898,  with  interest 
at  G^.     On  the  note  were  the  following  indorsements  : 

April  18,  1898,  $200 ;  July  18,  1898,  $250 ;  Sept.  18,  1898, 
$200. 
Find  amount  due  on  this  note  Jan,  18,  1899. 


IK^tEREST.  309 


l*ROBLEMS  IN  SIMPLE  INTEREST. 

457.  1.  At  6^  what  principal  will  yield  G^'  in  a  year  ?  60^ 
in  a  year  ?    $3  in  a  year  ?     $6  in  a  year  ?     $12  in  two  years  ? 

2.  What  is  the  rate  when  the  interest  of  $100  for  a  year  is 
$6  ?     When  it  is  $4  ?     When  it  is  $15  for  3  years  ? 

3.  In  what  time  will  $100  loaned  at  6^  yield  $6  interest  ? 
$18  interest  ? 

4.  In  what  time  will  $200  loaned  at  6^  yield  $6  interest  ? 
$12  interest  ? 

458.  In  Art.  442  we  found  that  the  interest  is  tlie  product 
of  three  factors,  expressed  in  the  equation 

iz=prt  (1) 

In  this  equation,  if  tliree  of  the  elements  are  given,  the 
other  can  be  found.  Dividing  both  members  of  (1)  by  rt, 
we  have 


^  =  r. 

(2) 

Dividing  both  members  of  (1)  hy  pt,  we 

have 

"fi 

(3) 

Dividing  both  members  of 

(1)  hypr,  we 

have 

t  =  i- 

pr 

(*) 

459.  Since  the  amount  equals  the  sum  of  the  principal 
and  interest,  we  have,  if  a  represents  the  amount, 

a  =  p  -\-  i,  or,  replacing  *  by  its  Ysdiieprt, 
a  =  p  +  prt,  or  a  =  p  {1  +  rt). 

Dividing  both  members  by  1  +  rt,  we  have 


310  SCHOOL  ARITHMETIC. 


WRITTEN    EXERCISES. 

460.  1.  What  principal  will  in  3  yr.  6  mo.  at  6^  produce 

$49.14  interest  ? 

(a) 
Interest  of  $x  for  3^  yr.  =  $49.14. 
Interest  of  $1  for  8^  yr.  =  $.21. 
.•.$x=  $49.14-=-  .21,  or  $234. 

(b) 
Using  equation  (2),  t  =  3i;  r  =  &%  =  M;  ^  =  $49.14. 

Hence,i?=.^^  =  $234. 

Find  the  principal  that  will  produce  : 

2.  $60  interest  in  2  yr.  at  6fo. 

3.  $125  interest  in  2  yr.  6  mo.  at  8fc. 

4.  $216  interest  in  8  months  at  6^. 

5.  $127.50  interest  in  3  yr.  6  mo.  15  da.  at  6^. 

6.  If  money  loaned  for  seven  months  at  4^^  produces  $210, 
how  much  is  loaned  ? 

7.  How  much  must  be  invested  at  5^  to  yield  $1500  interest 
quarterly  ? 

8.  If  I  receive  $2200   semi-annually  from  an  investment 
yielding  5^,  what  is  the  sum  invested  ? 

9.  At  what  rate  will  $234  produce  $49.14  interest  in  3  yr. 

6  mo.? 

(a) 

Interest  for  31  yr.  =  $49.14. 

Interest  for  1  yr.  =  $49.14  h-  3^  =  $14.04. 

.-.  the  rate  =  $14.04  -i-  $234  =  .06,  or  Q%. 

(b) 

Using  equation  (3),  i  =  $49.14;  p  =  $234;  t  =  3^. 

Hence,  r  =     ^^^^'^^      =  .06,  or  Q%. 

$284  X  3i 

Find  the  rate  when  the  interest  : 


10.  On  $325  for  1  yr.  6  mo.  is  $19.50. 

11.  On  $1400  for  3  yr.  9  mo.  is  $315. 

12.  On  $2500  for  2  yr.  10  mo.  is  $283.33^. 


INTEREST.  311 

13.  On  $1576  for   1  yr.  5  mo.  18  da.  is  $92.45. 

14.  In  wlmt  time  will  $234  yield  $40.14  interest  at  6^  ? 

(a) 
Interest  for  x  years  =  $49.14. 
Interest  for  1  year  =  $14.04. 
.-.  the  time  =  |49.14  -^  |14.04  =  3.5,  or  3|  yr. 

(b) 
Using  equation  (4),  i  =  $49.14;  j)  =  $234;  r  =  .06, 

Hence,  t  =  _  ^f^A^^    =  3.5,  or  3^  years. 
$234  X  .06 

In  what  time  will  : 

15.  $300  produce  $37.50  at  5^  ? 

16.  $G8o  produce  $123.30  at  4^. 

17.  $4000  produce  $1000  at  UJ  ? 

18.  What  principal  will  in  3  yr.  6  mo.  amount  to  $283.14, 

at  6^  ? 

(a) 
Amount  of  %x  for  3i  yr.  =  $283.14. 
Amount  of  $1  for  3^  yr.  =  $1.21. 
,'.%x  =  $283.14  H-  1.21  =  $234. 

(b) 
Using  equation  (5),  a  =  $283.14;  r  =  .06;'  ^  =  3^; 
1  +  ,.^  ^  1  +  .06  X  3i  ■=  1  +  .21  =  1.21. 

Hence,  p  =  t???-^  =  $234. 
1.21 

Find  the  principal  that  will  amount  to  : 

19.  $936  in  5  years  at  6^. 

20.  $843.60  in  8  years  at  6^. 

21.  $1844.40  in  222  days  at  4^. 

22.  $681.40  in  7  mo.  24  da.  at  5fc. 

23.  At  what  rate  will  $240  gain  $8.96  in  6  mo.  12  da.? 

24.  If  $1200  amounts  to  $1391  in  2  yr.  7  mo.  25  da.,  what 
is  the  rate  per  annum  ? 

25.  A  borrowed  $700  at  6^,  and  paid  in  full  $724.50.  How 
long  did  he  have  the  money  ? 


812  SCHOOL  ARITHMETIC. 

^6.  In  what  time  will  1100  amount  to  1200,  or  double  \tseU, 
at  6^? 

27.  In  what  time  will  $100,  at  6^,  gain  $100  ? 

28.  How  much  must  be  invested  at  7^  to  give  a  semi-annual 
income  of  $875  ? 

29.  What  principal  at  interest  at  6^  will  amount  to  $580.72 
in  7  mo.  12  da.? 

30.  What  sum  at  Qfo  will  amount  to  $795  in  a  year  ? 

31.  I  owe  $1302  which  is  to  be  j^aid  in  a  year.  What  cash 
payment  to-day  would  pay  the  debt  if  money  is  worth  5^  ? 

32.  A  house  is  offered  to  me  for  $2400  cash,  or  for  $2800 
if  not  paid  for  15  months.  If  money  is  worth  4J,^,  how  much 
better  for  me  is  the  cash  offer  ? 

ANNUAL.    INTEREST. 

461.  Simple  interest  on  the  principal  and  on  each  year's 
unpaid  interest  is  called  Annual  Interest. 

Notes  bearing  annual  interest  contain  the  words  "  interest  payable 
annually."     The  laws  of  some  of  the  states  do  not  allow  annual  interest. 

WRITTEN     EXERCISES. 

462.  1.  Find  the  amount  duo  at  maturity  on  a  note  for 

$400,  due  5  years  from  its  date,  with  interest  at  6^,  payable 

annually. 

Face  of  note  =  $400. 

Interest  on  $400  for  5  yr.  at  6%  =  $120. 

Interest  on  $24  for  4  yr.  +  3  yr.  +  2  yr.  +  1  yr.  at  6^  =  $14.40. 

$400  +  $120  -r  $14.40  =  $534.40,  the  amount. 

The  first  year's  interest  ($24)  draws  interest  4  yr.  :  the  second,  3  yr.  ; 
the  third,  2  yr.  ;  the  fourth,  1  yr.  This  is  the  same  as  one  year's  inter- 
est drawing  interest  10  years. 

Find  the  amount  at  annual  interest  of  ; 

2.  $1200  due  in  3  yr.  at  Qfo. 

3.  $500  due  in  4  yr.  6  mo.  at  6^. 

(Periods  =  3|  +  2i  +  li  +  i) 


INTEREST.  313 

4.  1350  due  in  3  yr.  8  mo.  15  da.  at  8^. 
6.  $840  due  in  3  yr.  at  5^. 

6.  $7000  due  in  4  yr.  at  G^. 

7.  AVhat  amount  is  due  June  20,  1898,  on  a  note  for  $350, 
dated  Jan.  5,  1895,  with  6^  interest,  payable  annually,  on 
wliicli  no  payments  have  been  made  ? 

8.  Find  tlie  amount  of  $1200  at  annual  interest  for  4  yr. 
at  6^.  Compare  the  amount  of  annual  interest  with  the 
amount  at  simple  interest. 

9.  Mr.  H  makes  a  note  for  $1000  for  3  yr.  3  mo.  with 
interest  at  6^  per  annum,  payable  semi-annually,  but  pays 
no  interest.     Find  the  amount  due  at  maturity. 

COMPOUND    INTEREST. 

463.  Interest  found  by  adding  unpaid  interest  to  the 
principal  at  stated  intervals  and  by  using  the  sum  as  a  new 
principal  is  called  Compouud  Interest. 

1.  When  the  interest  is  added  to  the  principal  at  the  end  of  each  year, 
the  interest  is  said  to  be  compounded  annually  ;  when  it  is  added  every 
three  months,  it  is  said  to  be  compounded  quarterly  ;  and  so  on. 

2.  Savings  banks  usually  allow  compound  interest  to  depositors,  and 
in  some  states  it  is  legal  on  funds  due  by  or  to  guardians.  It  is  also 
used  in  constructing  the  tables  of  bond  values  used  by  brokers  ;  other- 
wise it  is  not  in  general  use. 

WRITTEN     EXERCISES, 

464.  1.  What  is  the  compound  interest  on  $500  for  3 
years  at  5^  ? 

$500  +  interest  for  a  year      =  $535,  a  new  principal. 
$525  +       "         "  "        1=  $551.25,  " 

$551.25  +  interest  for  a  year  =  $578.81,  the  amount  due. 
$578.81  —  $500  =  $78.81,  the  compound  interest. 

2.  Find  the   compound   interest   on   $250   for  2   years   6 
months  at  Qfo,  interest  compounded  semi-annually. 
(The  interest  is  to  be  added  every  six  months.) 


314  SCHOOL  ARITHMETIC. 

3.  Find  the  compound  interest  on  $800  for  3  yr.  6  mo.  24 
da.  at  G^. 

The  amount  at  compound  interest  for  3  yr.  =  $952.81. 
Interest  on  $952.81  for  G  mo.  24  da.  =      32.40. 

Amount  for  3  yr.  6  mo.  24  da.  =  $985.21. 

$985.21  —  $800  =  $185.21,  the  compound  interest  required. 

Find  the  compound  interest  on  : 

4.  $360  for  3  yr.  6  mo.  at  5^. 

5.  $243.12  for  3  yr.  at  4^. 

6.  $300  for  2  yr.  4  mo.  at  6fc. 

7.  $500  for  2  yr.  4  mo.  at  (jfo,  compounded  semi-annually. 

8.  $735.60  for  2  yr.  5  mo.  24  da.  at  4^,  compounded  quarterly. 

9.  A  boy  deposits  $200  in  a  savings  bank  whicli  allows  4^ 
compound  interest,  and  adds  the  interest  to  the  principal 
every  6  months.  At  the  end  of  5  years  how  much  will  the 
bank  owe  tlie  boy  ? 

10.  On  the  first  day  of  June  each  year  a  man  deposits  $300 
in  a  savings  bank  which  allows  4:fo  interest,  compounded  semi- 
annually. How  much  will  the  bank  owe  the  man  at  the  ex- 
piration of  5  yr.  3  mo.  ? 

465.  The  compound-interest  table  on  the  opposite  page 
is  such  as  is  used  by  savings  banks  in  computing  compound 
interest,  and  by  investors  and  others  who  wish  to  compute 
the  amount  resulting  from  the  reinvestment  of  interest  as  it 
becomes  due. 

Find  the  compound  interest  of  the  following,  making  use 
of  the  table  : 

1.  $420  for  8  years  at  4^^. 

Amount  of  $1  for  8  yr.  at  ^%  =  $1.42210. 
Amount  of  $420  =  420  x  $1.42210  =  $597.28. 

2.  $510  for  5  yr.  6  mo.  at  5^. 

3.  $2500  for  4  yr.  3  mo.  at  Ifc. 

4.  $1050  for  2  yr.  8  mo.  at  6^. 

5.  $800  for  3  yr.  7  mo.  15  da.  at  3^. 

6.  $260.75  for  4  yr.  8  mo.  10  da.  at  3^^. 


INTEREST. 


315 


Table. 
Showing  the  amount  of  $1  at  compound  interest  for  : 


YR. 

2  PER  CENT. 

2i  PER  CENT. 

3  PER  CENT. 

3J  PER  CENT. 

4  PER  CENT. 

1 

1.02000 

1.0-3500 

1.03000 

1.03500 

1.04000 

2 

1.04040 

1.05063 

1.06090 

1.07123 

1.08160 

3 

1.06121 

1.07689 

1.09273 

1.10872 

1.12486 

4 

1.08243 

1.10381 

1 . 12551 

1.14752 

1.16986 

5 

1.10408 

1.13141 

1.15927 

1.18769 

1.21665 

6 

1.12616 

1.15969 

1 . 19405 

1.22926 

1.26532 

7 

1 . 14869 

1.18869 

1.22987 

1.27228 

1.31593 

8 

1.17100 

1.21840 

1.20677 

1.31681 

1.3(5857 

9 

1.19509 

1.24886 

1.30477 

1.36290 

1.423:U 

10 

1.21899 

1.28009 

1.34392 

1.41060 

1.48024 

11 

1.24337 

1.31209 

1.38423 

1.45997 

1.53945 

13 

1.26824 

1.34489 

1.42576 

1.51107 

1.60103 

13 

1.29361 

1.37851 

1.46853 

1.56396 

1.06507 

14 

1.31948 

1.41297 

1.51259 

1.61870 

1.73168 

15 

1.34587 

1.44830 

1.55797 

1.67535 

1.80094 

16 

1.37279 

1.484bl 

1.60471 

1.73399 

1 , 87298 

17 

1.40024 

1.52162 

1.65285 

1.79408 

1.94790 

18 

1.42825 

1.55966 

1.70243 

1.85749 

2.02582 

19 

1.45681 

1.59865 

1.75351 

1.92250 

2.10685 

20 

1.48595 

1.63862 

1.80611. 

1.98979 

2.19112 

YR. 

4i  PER  CENT. 

5  PER  CENT. 

h\  PER  CENT. 

C  PER  CENT. 

7  PER  CENT. 

1 

1.04500 

1.05000 

1.05500 

1.06000 

1.07000 

2 

1.09203 

1.10250 

1.11303 

1.12360 

1.14490 

3 

1.14117 

1.15763 

1.17424 

1.19102 

1.22504 

4 

1.19252 

1  21551 

1.28882 

1.26248 

1.31080 

5 

1.24618 

1.27628 

1.30696 

1.33823 

1.40255 

6 

1.30226 

1.34010 

1.37884 

1.41852 

1.50073 

7 

1.36086 

1.40710 

1.45468 

1.50363 

1.60578 

8 

1.42210 

1.47746 

1.53469 

1.59385 

1.71819 

9 

1.48610 

1.55133 

1.61909 

1.68948 

1.83846 

10 

1.55297 

1.62889 

1.70814 

1.79085 

1.96715 

11 

1.62285 

1.71034 

1.80209 

1.89830 

2.10485 

12 

1.69588 

1.79586 

1.90121 

2.01220 

2  25219 

13 

1.77220 

1.88565 

2.00577 

2.13293 

2.40985 

14 

1.85194 

1.97993 

2  11609 

2.26090 

2.57853 

15 

1.93528 

2.07893 

2.23248 

2.39656 

2.75903 

16 

2.02237 

2.18287 

2.35526 

2.54035 

2.95216 

17 

2.11338 

2.29202 

2.48480 

2.69277 

3.15882 

18 

2.20848 

2.40662 

2.62147 

2.85434 

3.37993 

19 

2.30786 

2.52695 

2.76565 

3.02560 

3.61653 

20 

2.41171 

2.65330 

2.91776 

3.20714 

3.86968 

BANKS  AND    iBANK    DISCOUNT. 

466.  A  Bank  is  an  institution  whose  ordinary  business  is 
to  receive  deposits  of  money,  to  make  loans,  to  discount  notes, 
and  to  sell  and  collect  drafts. 

467.  A  Check  is  the  written  order  of  a  depositor,  directing 
a  bank  to  pay  a  specified  sum  of  money  to  a  certain  person, 
or  to  his  order. 

The  following  is  a  common  form  : 


No,  iSb. 

Pensacola,  Fla.,    SHBaxck  i5,  ipoo. 

Ifirat  IHational  Banh  of  peneacola* 

Pay  to  the  order  of    ^.  (S^.  IJomA 

(Seventy. five  and   J^    Dollar S. 


Note. — If  a  check  is  made  payable  to  the  order  of  the  payee,  as  above, 
the  payee  must  indorse  it. 

468.  A  certified  check  is  a  check  upon  the  face  of  which 
the  cashier  of  the  bank  has  stamped  the  word  '^  Certified," 
with  date  and  name  of  bank,  and  has  written  his  signature  as 
cashier.     The  bank  is  then  responsible  for  its  payment. 

469.  Bank  Discount  is  the  sum  retained  by  a  bank  in 
cashing  a  note  before  it  is  due,  or  on  which  it  loans  money. 
It  is  simple  interest  on  the  maturity  value  of  the  note. 


BANKS  AND  BANK  DISCOUNT.  317 

470.  The  Proceeds  of  a  note  is  its  maturity  value  legs 
the  discount.  It  is  the  sum.  paid  out  by  a  bank  for  or  on  the 
note. 

471.  The  Day  of  Maturity  is  the  day  on  which  the  note 
becomes  legally  due  and  payable. 

Banks  do  no  business  on  Sundays  or  legal  holidays,  hence  notes 
falling  due  then  are  payable  on  the  succeeding  or  the  preceding  business 
day,  usually  the  former.  Find  out  whether  in  your  locality  notes 
falling  due  on  Saturday  are  payable  on  that  day  or  on  the  next  suc- 
ceeding business  day. 

472.  The  Term  of  Discount  is  the  time  for  which  the 
bank  computes  discount — the  period  for  which  it  charges 
interest.  It  begins  on  the  day  of  discount  and  ends  on  the 
day  of  maturity. 

Some  bankers  include  the  day  of  discount  in  the  time  for 
which  they  take  interest. 

473.  In  states  where  days  of  grace  have  not  been  abolished, 
three  days  are  allowed  in  addition  to  the  time  stated  in  the 
note  before  the  note  is  legally  due.     (See  Art.  449.) 

474.  A  Protest  is  a  notice  in  writing  by  a  notary  puMic 
to  the  indorsers  that  a  note  has  not  been  paid  on  the  day 
of  maturity.  If  a  note  is  not  protested  before  the  day  of 
maturity  ends,  the  indorsers  are  released  from  their  obli- 
gation. 

475.  In  finding  maturity,  banks  generally  count  forward 
by  months  or  by  days,  as  the  note  specifies.  The  term  of 
discount  is  usually  found  in  days.  In  the  examples  given  in 
this  book,  this  practice  is  observed. 

MAKING  LOANS. 

476.  A  person  wishing  to  borrow  money  from  a  bank 
nsually  makes  a  note  payable  to  the  order  of  some  person  who 
is  willing  to  indorse  it. 


318  SCHOOL  ARITHMETIC. 

The  following  is  a  common  form  of  such  note  : 


$100.  Richmondy  Va.,  BVoaxck  ib,  i^os. 

^wo   months    after  date  I  promise  to  pay  to 

the  order  of      cfblckaxd  cR)oe 

One  BSundxcd . —  Dollars 

at      %kc  cf'ixAi  National  €Bank,   d\>ichmond,    %, 

Value  received. 

Due, J'o/m  3)oe. 


When  Richard  Roe  indorses  this  note,  John  Doe  takes  it 
to  the  bank,  which  loans  him  1100  less  the  interest  (dis- 
count) for  61  days  at  the  legal  rate.  This  discount  is  $1.02, 
hence  Mr.  Doe  gets  198.98  ;  but  in  two  months  he  must  pay 
the  bank  1100. 

1.  By  indorsing  the  note  Richard  Roe  binds  himself  to  pay  it  if  John 
Doe  does  not  ;  and  since  he  indorses  tiie  note  to  enable  the  latter 
to  secure  the  loan,  that  is,  to  accommodate  the  maker,  he  is  called  an 
accommodation  indorser. 

2.  If  this  note  is  not  paid  May  15,  it  goes  to  protest. 

3.  No  interest  is  specified  in  notes  of  this  kind,  but  the  legal  rate  is 
charged. 

WRITTEN     EXERCISES. 

477.  1. 

$1000.  New  York,  Dec.  31,  1900. 

Two  months  after  date  I  promise  to  pay  J.  A.  Greene,  or 
order,  one  thousand  dollars,  value  received. 

J.    B.    MUKDOCK. 

This  note  was  discounted  Dec.  31  at  6^,  the  legal  rate. 
Find  the  proceeds. 


BANKS   AND  BANK  DISCOUNT.  319 

The  date  of  maturity  =  Feb.  28,  1901. 
The  term  of  discount  =  59  days. 

The  discount  =  interest  on  $1000  for  59  da.  =  $9.83. 

The  proceeds  =  $1000  -  $9.83  =  $990.17. 

j^OTE. — In  Pennsylvania,  and  in  other  states  where  the  day  of  discount 
is  included,  the  term  of  discount  in  the  above  note  would  be  60  days  ; 
where  grace  is  allowed,  it  would  be  3  days  more.  In  Boston,  and  in 
some  other  places,  when  the  time  a  note  has  to  run  is  expressed  in  months, 
the  term  of  discount  is  computed  for  this  number  of  months,  and  not 
for  the  exact  number  of  days. 


$000.  Philadelphia,  Pa.,  Feb.  28,  1898. 

Three  months  after  date  I  promise  to  pay  to  William  Post, 
or  order,  six  hundred  dollars,  value  received. 

W.  E.  Sankey. 

Discounted  Feb.  28.     Find  proceeds. 

Day  of  maturity  =  May  31  (the  28th  being  Saturday,  and  the  30th  a 
legal  holiday).     Term  of  discount  =  93  days. 

3. 

14000.  Richmond,  Va.,  June  30,  1899. 

Sixty  days  after  date  I  promise  to  pay  to  J.  F.  Guffey,  or 
order,  four  thousand  dollars,  value  received. 

W.  R.  Ford. 

Discounted  June  30,  at  6^.     Find  proceeds. 

4. 

$2500.  Columbia,  S.  C,  Aug.  31,  1901. 

Three  months  after  date  I  promise  to  pay  to  the  order  of 
J.  F.  Bunn,  two  thousand  five  hundred  dollars,  value  re- 
ceived. 

AV.  H.  McKelvey. 

Discounted  Aug,  31^  at  6^.  Find  the  proceeds.  (Grace 
allowed.) 


320  SCHOOL  ARITHMETIC. 

5. 

$3000.  Hartford,  Conn.,  May  4,  1903. 

Sixty  days  after  date  I  promise  to  pay  J.  M.  Clark,  or 
order,  three  thousand  dollars,  value  received. 

R.  J.  Stoney,  Jr. 
Discounted  May  4. 

6. 

$750.  Frankfort,  Ky.,  April  7,  1900. 

Three  months  after  date  I  promise  to  pay  to  the  order  of 
W.  M.  Gill,  seven  hundred  fifty  dollars,  value  rec'd. 

Geo.  H.  Welshons. 

Discounted  April  7.     (Grace  allowed,  and  day  of  discount 
included.) 

7. 
$1500.  Baltimore,  Md.,  March  30,  1902. 

Two  months  after  date  I  promise  to  pay  to  Howard  Welsh, 
or  order,  one  thousand  five  hundred  dollars,  value  rec'd. 

W.  S.  Finney. 
Discounted  Mar.  30.     (Day  of  discount  included.) 

8. 

$1000.  Wilmington,  Del.,  May  18,  1901. 

Sixty  days  after  date  I  promise  to  pay  William  Pollock,  or 
order,  one  thousand  dollars,  value  received. 

H.  W.  Walker. 

Discounted  May  18. 

9. 

$500.  Charlotte,  N.  C,  April  15,  1900. 

Three  months  after  date  I  promise  to  pay  to  the  order  of 
Anna  Bamford,  five  hundred  dollars,  value  received. 

George  Dewey. 

Discounted  April  15.     (Grace  allowed.) 


BANKS  AND  BANK  DISCOUNT.  321 

10. 

$10,000.  Savannah,  Ga.,  May  25,  1901. 

Two  months  after  date  I  promise  to  pay  to  the  order  of 
W.  H.  McCleary,  ten  tliousand  dollars,  value  received. 

John  D.  Miller. 

Discounted  May  25.     (Grace  allowed.) 

11. 

$400.  Nashville,  Tenn.,  May  4,  1900. 

Sixty  days  after  date  I  promise  to  pay  to  Wm.  H.  McGary, 
or  order,  four  hundred  dollars,  value  received. 

Samuel  Harper. 

Discounted  May  4.     (Grace  allowed.) 

12. 

$2000.  Austin,  Tex.,  Jan.  15,  1901. 

Sixty  days  after  date  I  promise  to  pay  to  the  order  of 
H.  M.  Jones,  two  thousand  dollars,  value  received. 

W.  W.  Ulerich. 

Discounted  Jan.  15,  1901.     (Grace  allowed.) 

13. 

$6000.  Little  Rock,  Ark.,  May  9,  1901. 

Ninety  days  after  date  I  promise  to  pay  to  the  order  of 
R.  S.  Latham,  six  thousand  dollars,  value  received. 

J.  D.  Anderson. 

Discounted  May  9,  1901.     (Grace  allowed.) 

DISCOUNTING  NOTES. 

478.  John  Mason  bought  a  lot  from  Richard  Adams  for 
$2000,  but  not  having  the  ready  money  agreed  to  pay  the 
$2000  in  sixty  days,  together  with  interest  at  6^. 

He  gave  the  following  note  : 
21 


322  SCHOOL  ARITHMETIC. 

$2000.  ^^Ew  York,  March  16,  1900. 

Sixty  days  after  date.,  for  value  received,  I  promise  to  pay 
to  the  order  of  Richard  Adams,  two  thousand  dollars,  with 
interest  at  6^.  JoHN"  Mason. 

Needing  money,  Richard  Adams  took  the  note  to  a  bank 
and  had  it  discounted  the  same  day,  March  16,  transferring 
the  note  to  the  bank  by  indorsement.  The  bank  paid  to 
Mr.  Adams  the  maturity  value,  less  Gfo  interest  thereon  for 
60  days. 

1.  The  note  matures  May  15,  at  which  time  John  Mason  must  pay 
the  bank  $2020,  the  maturity  value.  The  bank  retains  the  interest  on 
this  at  6%  for  60  days,  or  $20.20  ;  hence  the  owner  of  the  note,  Mr. 
Adams,  gets  $2020  -  $20.20  =  $1999.80. 

3.  So  far  as  the  bank  is  concerned,  this  process  of  discounting  notes 
purchased  by  way  of  discount — being  in  effect  a  mode  of  lending  money 
— is  essentially  the  same  as  that  of  lending  money  on  an  indorsed 
promissory  note  ;  but  there  is  the  important  distinction  that  the  indorser 
is  not  now  an  accommodation  indorser  ;  he  is  the  oivner  of  the  note  and 
is  the  party  who  receives  the  money  from  the  bank. 

3.  Notice,  also,  that  the  note  in  the  latter  case  usually  draws  interest, 
and  is  frequently  not  discounted  on  the  day  of  its  making.  When  it  is 
discounted  at  a  subsequent  date,  the  discount  is  reckoned  on  the  maturity 
value  for  the  time  from  the  day  of  discount  to  the  day  of  maturity. 

,    _  WRITTEN     EXERCISES. 

479.  1. 

$800.  Richmond,  Va,  June  17,  1900. 

Three  months  after  date  I  promise  to  pay  to  the  order  of 
L.  F.  Graham,  eight  hundred  dollars,  value  received,  at  the 
First  National  Bank,  with  interest  at  6^.     Jno.  B.  Head. 

Discounted  July  12,  at  6^.     Find  the  proceeds. 

Date  of  maturity  =  Sept.  17. 

Maturity  value     =  $800  +  interest  for  3  mo.==  $812. 

Term  of  discount  =  67  days  {i.e.,  19  in  July,  31  in  Aug.,  17  in  Sept.). 

The  discount        =  the  interest  on  $812  for  67  days  at  6%  ~  $9.07. 

The  proceeds        =  $812  -  $9.07  =  $802.93. 

(Many  banks  in  Virginia  charge  for  day  of  discount.) 


BANKS  AND  BANK  DISCOUNT.  323 

2. 

$300.  Pittsburg,  Pa.,  July  10,  1900. 

Sixty  (lays  after  date  I  promise  to  pay  to  J.  F.  Miller,  or 
order,  three  hundred  dollars,  with  interest  at  0^,  value 
received.  Lewis  Moran. 

Discounted  Aug.  6,  at  6^.  Find  the  proceeds.  (Day  of 
discount  included.) 

3. 

1500.  Dayton,  0.,  Sept.  29,  1901. 

Two  montlis  after  date  I  promise  to  pay  to  James  II.  Piatt, 
or  order,  five  hundred  dollars,  value  received. 

C.   IIORNUNG. 

Discounted  Sept.  29,  at  iWc.     Find  proceeds. 

4. 

$1000.  Jackson,  Miss.,  Dec.  31,  1900. 

Two  months  after  date  I  promise  to  pay  to  Thomas  E.  Boyd, 
or  order,  one  thousand  dollars,  witli  interest  at  (j^/>,  value 
received.  S.  C.  Hepler. 

Discounted  Dec.  31,  at  6i.  Find  proceeds.  (Grace 
allowed.) 

5. 

James  11.  Ljifferty  wishing  to  borrow  some  money  for  two 
montlis  from  the  Atlas  National  Bank  of  Boston  gives  a 
promissory  note  for  11800,  George  Gosser  being  the  accom- 
modation indorser.  Draw  the  note,  dating  it  May  4,  1900, 
and  find  the  proceeds. 

6. 

Prepare  a  90-day  note  for  $250  on  which  you  can  obtain  a 
loan  from  a  bank  in  your  locality.  Compute  the  bank's 
charge  for  discounting  it  in  accordance  with  local  practice. 

7.      , 
1  have  a  note  for  $5000  dated  June  5,  and  payable  three 
montlis  after  date,  with  6^  interest.     How  much  would  a 
bank  in  your  locality  give  me  for  the  note  on  July  3  ? 


324  SCHOOL  ARITHMETIC. 

8. 

For  what  sum  is  a  60-day  note  given  wlien  a  bank  dis- 
counting it  at  8^  gives  the  maker  $725,  allowing  days  of 
grace  ? 

The  discount  on  |1  for  63  days  at  8^  -  |.014. 
The  proceeds  of  a  $1  note  =  $1  -  $.014  —  $.986. 
The  face  required  =  $725  -f-  .986  =  $735.29. 

9. 

If  yon  wish  to  procure  $1000  from  a  bank  for  ninety  days 
at  6^,  for  what  sum  must  you  write  the  note  ? 

10. 

The  National  Union  Bank  of  New  York  (which  counts 
neither  grace  nor  day  of  discount)  loans  J.  0.  Brown  $7500 
on  a  3-month  note  dated  April  4,  J.  W.  Lee  being  the 
indorser.     Write  the  note. 


PRESENT    WORTH    AND    TRUE    DISCOUNT. 

480.  The  Present  Worth  of  a  debt  is  the  sum  which,  if 
put  at  simple  interest,  would  amount  to  tlie  debt  when  due. 

481.  The  difference  between  the  amount  of  the  debt  and 
its  present  worth  is  called  the  True  Discount. 

WRITTEN    EXERCISES. 

482.  1.  Find  the  present  worth  and  true  discount  of  $6^1 
due  in  2  yr.  6  mo.,  if  money  is  worth  6^. 

(a) 
Amount  of  $x  for  2i  yr.  =  $621. 
Amount  of  $1  for  2^  yr.  =  $1.15. 
^x  -  $621  -r-  1.15  =  $540,  the  present  worth. 
The  true  discount  =  $621  -  $540  =  $81. 

(b) 
In  equation   (5),  Art.  459,  a  ^  $621  ;  r  =  .06  ;  ^  =  2^  ;  1  +  r^  =  1.15. 
Hence  p  =  $621  -f-  1.15  =  $540. 


STOCKS   AND  BONDS.  325 

2.  Find  the  present  worth  of  $590  due  in  1  yr.  6  mo.,  the 
current  rate  of  interest  being  6^. 

3.  Find  the  true  discount  on  i\  debt  due  in  4  mo.  10  da., 
the  debt  being  $450,  and  money  being  worth  5^. 

4.  Find  the  present  worth  and  true  discount  of  $1235  due 
in  1  yr.  7  mo.  12  da.,  the  current  rate  being  6^. 

5.  A  man  buys  flour  for  $2840  on  six  months'  time.  If 
payment  is  made  at  the  time  of  purcliase,  how  much  should 
be  deducted  from  tlie  bill,  money  being  worth  6^? 

6.  Wluit  is  the  difference  between  the  true  discount  of 
$G40,  due  in  1  yr.  3  mo.  15  da.,  and  the  interest  on  the  same 
amount  for  the  same  time,  money  being  worth  Q^  ? 

7.  Wiiat  sum  must  I  put  at  interest  at  8^  to  liquidate  a 
debt  of  $1250  due  2  yr.  6  mo.  hence  ? 

8.  I  am  offered  $8250  cash  for  my  farm  by  one  man,  and 
another  offers  me  $8580  in  4  months,  without  interest. 
Which  offer  is  the  better,  if  money  is  worth  6^  ? 

■  9.  What  is  the  difference  between  the  bank  discount  and 
the  true  discount,  each  at  5i,  on  a  note  for  $654  due  in  90 
days  ?  Show  that  the  bank  discount  equals  the  true  dis- 
count plus  the  interest  on  the  true  discount. 

STOCKS    AND    BONDS. 

483.  When  a  numt?er  of  persons  wish  to  engage  in  any 
extensive  business,  they  usually  form  themselves  into  an 
association  called  a  Stock  Company. 

484.  The  sum  of  money  subscribed  by  the  members  of 
the  company  to  inaugurate  the  business  is  called  the  Capital 
Stock. 

485.  The  capital  stock  is  usually  divided  into  a  definite 
number  of  shares  of  a  specified  value,  and  is  issued  in  the 
form  of  certificates  of  stock,  each  stating  that  the  person 
named  therein  owns  so  many  shares, 


326  SCHOOL  ARITHxMETIC. 

486.  The  value  of  a  share  named  in  the  certificate  of 
stock  is  called  the  Par  Value.  The  stock  is  usually  divided 
into  shares  of  the  face  or  par  value  of  $100  each. 

1.  If  the  value  of  each  share  is  $100,  what  is  the  par  value 
of  10  shares  ?     100  shares  ? 

487.  The  price  at  which  stocks  are  selling  in  the  market 
is  called  their  Market  Value. 

1.  When  stock  sells  at  5^  above  par  value,  what  is  the 
market  value  of  a  $100  share  ? 

488.  A  stock  is  said  to  be  at  a  premium,  or  ahove  par, 
when  it  sells  for  more  than  its  face  value;  it  is  said  to  be  at  a 
discount,  or  below  par,  when  it  sells  for  less  than  its  face  value. 

Thus,  if  a  stock  is  quoted  at  107,  $100  stock  sells  for  $107,  and  the 
stock  is  at  1%  ijremium  ;  if  it  is  quoted  at  93,  $100  stock  sells  for  $93, 
and  the  stock  is  at  1%  discount. 

1.  If  stock  is  quoted  at  106,  what  is  the  rate  of  premium  ? 
Why  do  stocks  vary  in  price  ? 

489.  If  the  company  makes  more  than  its  expenses,  part 
or  all  of  the  surplus  is  divided  among  the  stockholders  as 
Dividends.  The  dividends  are  usually  expressed  as  a  cer- 
tain per  cent  of  the  par  value,  but  sometimes  as  a  certain 
number  of  dollars  a  share. 

1.  If  a  company  declares  a  dividend  of  10^,  how  much 
does  the  owner  of  a  $100  sliare  get  ? 

490.  A  written  obligation  under  seal  securing  the  pay- 
ment of  a  sum  of  money  at  a  specified  time,  and  bearing  a 
certain  rate  of  interest,  is  called  a  Bond. 

1.  Which  would  be  preferable  to  owu,  $1000  of  stock  in 
a  company,   or  one  of   its  $1000    bonds,   if   each  pays  5^? 

491.  Bonds  are  issued  by  governments  (local,  state,  or 
national),  or  by  stock  companies,  for  the  purpose  of  effecting 
loans.  They  are  of  two  kinds — registered  bonds  and  coupon 
bonds. 


STOCKS  AND  BONDS.  327 

492.  Registered  bonds  are  recorded  by  their  numbers  and 
the  names  of  the  persons  owning  them,  and  cannot  be  trans- 
ferred without  a  change  in  the  record  kept  by  tlie  party 
issuing  them.  Coupon  bonds  have  attached  to  them  coupons, 
or  certificates  of  interest,  which  are  detached  as  interest 
becomes  due,  and  presented  for  payment. 

493.  Bonds  may  be  bought  and  sold  in  the  market  in  tlie 
same  manner  as  stocks,  and  are  designated  in  quotations  by 
the  name  of  tlie  company  or  government  issuing  them,  and 
the  rate  of  interest  they  bear,  with  the  date  of  maturity,  and 
whether  registered  or  coupon. 

Thus,  "U.  S.  4's,  coup.,  1925"  means  United  States  coupon  bonds 
bearing  4  per  cent  interest,  the  principal  payable  in  1925. 

Note. — Bonds  pay  interest  on  their /ace  value  at  a  fixed  rate,  hence 
thkir  market  price  does  not  affect  tlie  interest  they  yield.  The  income 
from  stocks  is  variable,  as  it  depends  upon  the  prosperity  of  the  busi- 
ness. 

494.  Persons  who  buy  and  sell  stocks  and  bonds  are  called 
Stock  Brokers,  and  their  commission  is  called  Brokerage, 
The  brokerage  is  usually  ^fo  of  the  par  value.  This  is  cliarged 
for  huying  and  also  for  selling. 

1.  If  a  broker  charges  |^  for  selling  10  shares  of  stock, 
how  much  w^U  be  the  brokerage  ? 

495.  The  market  values  of  stocks  and  bonds  as  given 
daily  in  the  newspapers  are  called  Stock  Quotations. 

A  quotation  of  127  means  that  $100  of  stock  is  selling  for  $127.  In 
this  case  the  seller  receives  $127  —  $^,  or  $126|  for  each  share,  and  the 
buyer  pays  $127  +  $-«,  or  $127g,  for  each  share,  provided  the  deal  is 
made  through  a  broker. 

1.  What  will  a  seller  receive  from  his  broker  for  one  share 
of  stock  sold  at  11^,  brokerage  ^^o  ?     What  for  10  shares  ? 

2.  What  will  a  buyer  have  to  pay  for  one  share  of  stock 
purchased  at  97J,  brokerage  ^^  ?    What  for  10  shares  ? 


328  SCHOOL  ARITHMETIC. 

496.  The  following  are  quotations  of  U.  S.  bonds  in  the 
market  of  June  7,  1900  : 

BID.  ASKED. 

U.  S.  3's,  reg 109^  109| 

U.  S.  3's,  coup 109i  109f 

4'8,  reg.,  1907 114^  115 

4's.  coup.,  1907 115i  116 

4's,  reg.,  1925 134^  135 

4's,  coup.,  1925 134i  135 

5's,  reg.,  1904 113^  114 

5's,  coup.,  1904 113i  114 

497.  The  following  are  from  the  stock  quotations  of  the 
same  day  : 

STOCKS.  HIGHEST.  LOWEST.  CLOSING. 

Am.  Sugar  Ref 116i  114|  1141 

Am.  Tobacco  Co 128  128  128' 

Brooklyn  Rap.  Tr 69f  68f      68i 

C.  B.  &  Q 129i  128i  128i 

Del.  &  Hud 113^  113  113 

Ches.  &  Ohio 27f  27|  27f 

N.  Y.  Central 131i  130i  130^ 

Pennsylvania  R.R 130|  129|  128| 

Northern  Pacific 60  59|  59f 

Southern  Railway 12  12  12 

1.  What  would  I  receive  for  one  share  of  N.  Y.  Central  at 
the  highest  quotation,  brokerage  ^^  ?     What  for  10  shares  ? 

2.  What  would  one  share  of  Pennsylvania  R.R.  stock  cost 
at  the  closing  quotation,  brokerage  ^^  ?  10  shares  ?  How 
many  shares  could  be  bought  for  $2580  ? 

3.  What  is  the  difference  between  the  highest  and  the 
lowest  quotations  of  Brooklyn  Rapid  Transit  stock  on  the 
given  date  ? 

WRITTEN     EXERCISES. 

498.  1.  What  is  the  cost  of  84  shares  of  bank  stock  at  95^, 
brokerage  ^^  ? 

1  share  costs  $95^  +  $i  =  $951. 
84  shares  cost  84  x  |95i  =  $8011.50. 


STOCKS  AND  BONDS.  329 

2.  What  will  40  shares  of  Northern  Pacific  stock  cost  at 
59f,  brokerage  |  per  cent  ? 

3.  How  much  must  be  paid  for  125  shares  of  Delaware  and 
Hudson  stock  at  113,  brokerage  ^  per  cent  ? 

4.  My  broker  bought  for  me  75  shares  of  C.  B.  &  Q.  stock 
at  129^,  charging  ^fo  brokerage.     Find  the  cost. 

5.  How  many  shares  of  railroad  stock  at  104|-  can  be 
bought  for  $9450,  brokerage  ^  per  cent  ? 

1  share  costs  1104^  +  $i  =  $105. 

.-.  the  number  of  shares  =  $9450  -4-  $105  =  90. 

6.  How  many  shares  of  Brooklyn  Rapid  Transit  stock  can 
be  bought  for  $3475  if  the  quotation  is  69f ,  and  brokerage  |^  ? 

7.  When  Federal  Steel  is  quoted  at  33f ,  how  many  shares 
can  be  bought  for  $1355,  brokerage  ^  per  cent  ? 

8.  I  sent  my  broker  $7938  with  which  to  buy  Canadian  Pa- 
cific stock  at  94f,  brokerage  ^^.    How  many  shares  did  I  get  ? 

9.  What  annual  income  will  be  realized  from  $4982  in- 
vested in  4^  stock  at  105^,  brokerage  ^  per  cent  ? 

1  share  costs  $105^  +  $i  =  $106. 

The  number  of  shares  =  $4982  -f-  $106  =  47. 
.-.  the  income  =  47  x  $4  =  $188. 

10.  What  income  will  be  derived  from  $4565  invested  in 

railroad  stock   at   114,  brokerage  -^^,  if  the  stock  pays  6^ 
dividends  annually  ? 

11.  What  annual  income  will  a  man  receive  if  he  invests 
$12830  in  bank  stock  paying  quarterly  dividends  of  3^,  pro- 
vided the  stock  is  bought  at  160f,  and  no  brokerage  paid  ? 

12.  I  invested  $4535  in  U.  S.  4's  at  113^,  brokerage  i^. 
What  does  the  government  pay  me  annually  ? 

13.  What  sum  of  money  must  be  invested  in  3^  stock  at 
90,  brokerage  ^^,  to  realize  an  annual  income  of  $1800  ? 

1  share  yields  an  income  of  $3. 

The  number  of  shares  =  $1800  -^  $3  =  600. 

1  share  costs  $90  +  $i  =  |90i. 

600  shares  cost  600  x  $90^  =  $54075,  the  investment. 


330  SCHOOL  ARITHMETIC. 

14.  A  man  has  4^  U.  S.  bonds  on  which  the  quarterly 
interest  amounts  to  $480.  He  bought  the  bonds  at  105. 
How  much  did  he  pay  for  them  ? 

15.  A  certain  bank  pays  4^  semi-annual  dividends,  my 
share  of  which  amounts  to  $360.  If  the  stock  cost  me  152^ 
and  ^^  brokerage,  how  much  have  I  invested  in  that  se- 
curity ? 

16.  A  man  sold  144  shares  of  railroad  stock  at  2^^  below 
par,  paying  1%  brokerage.  With  the  proceeds  he  bought 
Cotton  Oil  stock  at  36f,  brokerage  ^fo.  How  many  shares 
did  he  get,  and   how  many  dollars  were  left  over  ? 

17.  If  250  shares  of  Southern  Railway  are  bought  at  12^ 
and  sold  at  13^,  brokerage  ^^  in  each  case,  what  is  the 
profit  ? 

18.  A  speculator  sold  through  his  broker  240  shares  of 
Am.  Sugar  stock  at  152f.  It  cost  him  $27720.  What  was 
his  gain? 

19.  Mr.  Jones  bought  75  shares  of  Am.  Tobacco  at  142^, 
held  it  a  year,  and  sold  it  at  210.  If  he  received  a  15^  divi- 
dend and  paid  ^<fo  brokerage  each  way,  what  was  his  total 
gain  ? 

20.  If  I  buy  100  shares  of  Delaware  and  Hudson  stock  at 
115  and  sell  them  6  months  later  at  112^,  after  receiv- 
ing a  3J^  dividend,  what  is  my  loss,  money  being  worth  4^  to 
me  ? 

21.  When  Missouri  Pacific  stock  is  quoted  at  47^  and  is 
paying  2|^^  dividends,  what  sum  invested  in  it  will  give  me 
an  income  of  $2000  a  year  ? 

22.  How  many  shares  of  Chesapeake  and  Ohio  stock  at  285- 
are  worth  as  much  as  50  shares  of  Air  Brake  at  228  ? 

23.  One  morning  a  man  bought  1000  shares  of  railroad 
stock  at  133^  ;  in  the  afternoon  he  sold  it  at  135.  What  was 
his  profit  after  paying  brokerage  of  ^<fo  each  way  ? 

24.  In  1893  I  bouglit  tliirty  h'fc  city  bonds,  par  value  $100, 
at  108J,  brokerage  ^fo.     They  fell  due  6  years  later,  when  the 


REVIEW   WORK.  331 

city  redeemed  them  (paid  me  their  par  value).     What  was 
my  net  profit  ? 

25.  Jan.  3,  1890,  Henry  Howard  bought  220  shares  of  rail- 
road stock  at  92§,  which  paid  liim  dividends  as  follows  :  in 
1890,  4^  ;  in  '91,  4^^  ;  in  '92,  3^  ;  in  '93,  none;  in  '94,  Ij^  ;  in 
'95,  2^^  ;  in  '9G,  4^  ;  in  '97,  H.  Jan.  3,  1898,  he  sold  the  stock 
at  118|.  How  much  would  he  have  lost  or  gained  by  lending 
his  money  at  6^  interest  ? 

26.  Mr.  C.  C.  Davis  bought  80  shares  of  bank  stock  at 
149 J,  brokerage  ^^.  If  his  investment  yielded  him  10;^, 
what  quarterly  dividends  did  the  bank  declare  ? 

27.  What  per  cent  on  my  investment  will  I  receive  if  I 
buy  5fo  bonds,  interest  payable  annually,  at  791,  brokerage 
^^,  not  considering  the  length  of  time  the  bonds  run  ? 

Each  $80  invested  yields  |5. 

.'.  the  rate  of  income  =  g^o,  or  6J^. 

28.  Including  ^^  brokerage,  how  much  must  I  pay  for 
4^  bonds,  interest  payable  annually,  so  that  I  may  realize  an 
income  of  5fo  on  the  investment,  not  considering  the  length 
of  time  the  bonds  run  ? 

,  5%  of  the  cost  of  each  bond  (|100)  =  $4. 
100,^''  .  "  "  "  "  "  =$80 
(The  quotation  =  80  -  i  =  79|.) 


REVIEW    WORK. 

ORAL  EXERCISES. 

499.  1.  What  is  tlie  interest  on  $200  for  5  yr.  G  mo.  at  6 

per  cent  ? 

2.  If  I  lend  my  money  at  5^,  and  receive  1100  a  year  in- 
terest, how  much  have  I  loaned  ? 

3.  Five  per  cent  of  William's  money  is  $1  more  than  4^ 
of  it.  Should  he  loan  it  at  6fo,  in  how  many  years  would  the 
interest  amount  to  $50  ? 


332  SCHOOL  ARITHMETIC. 

4.  In  how  many  years  will  one  dollar  amount  to  three  dol- 
lars, if  money  is  worth  10^  ? 

6.  If  the  interest  I  pay  at  6^  is  a  dollar  a  month,  how 
much  have  I  borrowed  ? 

6.  A  man  whose  salary  is  $2000  a  year  spends  70^  of  it. 
How  much  does  he  save  a  month  ? 

7.  If  an  organ  costs  $60,  at  what  price  must  it  be  sold  to 
gain  60  per  cent  ? 

8.  Hart  spent  40^  of  his  money  for  a  cart,  50^  for  a  watch, 
and  the  remainder,  which  was  $5,  for  a  hat.  How  much  had 
he  at  first  ? 

9.  After  paying  out  60^  of  his  money  for  rent,  and  60^  of 
the  remiiinder  for  coal,  Mr.  A  had  $32  left.  How  much  did 
he  pay  for  rent  ? 

10.  When  the  interest  at  8^  is  f  of  the  principal,  how 
many  years^  interest  is  due  ? 

11.  In  how  many  years  will  $5  at  5^  amount  to  $15  ? 

12.  The  width  of  a  board  is  10^  of  its  length,  and  the 
perimeter  of  the  board  is  22  feet.     What  is  its  length  ? 

13.  Mr.  A  received  $80  for  selling  $200  worth  of  books. 
What  rate  of  commission  did  he  receive  ? 

14.  When  goods  that  cost  30  cents  a  yard  are  sold  at  37^ 
cents  a  yard,  what  per  cent  is  gained  ? 

15.  When  bicycles  are  marked  down  in  price  from  $75  to 
$50,  by  what  per  cent  is  the  price  reduced  ? 

16.  One  eighth  of  an  acre  is  what  per  cent  of  half  an 
acre  ? 

17.  A  real  estate  agent  sold  a  lot  for  me,  and,  retaining 
his  commission  of  3^,  remitted  $291.  For  what  sum  did  he 
sell  the  lot  ? 

WRITTEN     EXERCISES. 

500.  1.  What  is  the  amount  of  $1000  for  7  yr.  10  mo.  18 
da.  at  Q^  simple  interest  ? 

2.  Find  the  difference  between  the  simple  interest  and  the 
compound  interest  of  $1200  for  3  years  at  6  per  cent. 


REVIEW  WORK. 


3.  How  much  does  the  annual  interest  exceed  the  simple 
interest  of  $400  for  2^  years  at  5  per  cent  ? 

4.  I  paid  I1G5.375  for  the  use  of  $1350  at  7^.  How  long 
did  I  have  it  ? 

5.  What  principal  at  C^  will  amount  to  $139.86  in  7 
months  and  6  days  ? 

6.  A  note  for  $700  due  in  3  months-,  without  interest,  is 
purchased  by  a  broker  for  $650.  What  rate  of  interest  does 
he  receive  ? 

7.  What  sum  at  simple  interest  for  5  years  at  5^  will 
amount  to  $1075.50? 

8.  A  man  loaned  a  neighbor  a  sum  of  money  at  4^^  interest. 
At  the  end  of  18  months  the  debt  was  paid  in  full  by  a  check 
for  $1814.75.  How  much  more  than  the  sum  borrowed  did 
he  return  ? 

9.  Mr.  A  bought  a  horse  for  $124.80,  to  be  paid  in  8 
months  without  interest,  and  sold  him  at  once  for  $180. 
What  was  his  gain  per  cent,  money  being  worth  6  per  cent  ? 

10.  What  is  the  interest  on  a  sum  of  money  that  amounts 
to  $653.48  in  243  days  at  8  per  cent  ? 

11.  In  1  year  4  months  $311.50  amounted  to  $348.88,  at 
simple  interest.     What  was  the  rate  ? 

12.  A  man  bought  a  horse  in  New  York  and  paid  $20  for 
his  transportation  into  Canada,  where  he  sold  him  at  a  loss 
of  $13.  Had  he  paid  no  transportation  cliarges,  he  would 
have  gained  3^^.     How  much  was  paid  for  the  horse  ? 

13.  After  spending  25^  of  his  money,  and  25^  of  the  re- 
mainder, Harry  had  $731.25  left.  How  much  had  he  at 
first? 

14.  If  I  of  the  selling  price  is  gain,  what  is  the  per  cent  of 
profit  ?  . 

15.  What  sum  of  money  compounded  semi-annually  at  6^ 
will  amount  to  $2500  in  10  years  ? 

16.  What  is  the  present  worth  and  true  discount  of  a  debt 
of  $1000  due  in  1  year  6  months,  the  rate  being  6^  ? 


334  SCHOOL  ARITHMETIC. 

17.  What  would  be  the  annual  income  from  an  investment 
of  $8250  in  0^  stocks  at  82f ,  brokera<^e  ^  per  cent  ? 

18.  If  5^  stock  is  purchased  at  125,  what  per  cent  of  the 
investment  is  the  income  ? 

19.  For  how  much  must  I  give  my  note  at  bank  to  obtain 
$1000  for  90  days,  the  rate  of  discount  being  6  per  cent  ? 

20.  A  boy  sold  two  knives  at  the  same  price,  gaining  20^ 
on  one,  and  losing  20^  on  the  other.  He  lost  2  cents  by 
the  transaction.    What  did  each  cost  ? 

21.  An  agent  sells  a  typewriter  for  1100,  taking  a  note  to 
be  paid  in  10  equal  monthly  joayments  without  interest. 
After  half  the  payments  have  been  made  he  takes  the  note  to 
bank  and  has  it  discovmted  at  G^.     Find  the  proceeds. 


22.  Mr.  Henry  bought  land  at  $30  an  acre.  How  much 
must  he  ask  an  acre  in  order  that  he  may  reduce  his  asking 
price  25^  and  yet  make  20^  on  the  purchase  money  ? 

23.  A  man  invested  a  certain  amount  and  sold  at  a  loss  of 
30^.  He  invested  the  proceeds  and  sold  at  a  gain  of  30^. 
W^hat  per  cent  did  he  lose  on  the  two  speculations  ? 

24.. By  selling  a  hat  at  an  advance  of  IG^fo,  a  merchant 
gained  50  cents.  What  did  a  dozen  such  hats  cost  the  mer- 
chant ? 

25.  A  dealer  sold  two  books  for  $1  each.  On  one  he  made 
100^,  on  the  other  he  lost  50^.     AVhat  did  each  book  cost  ? 

■26.  Allowing  10;^  for  delinquent  taxes,  and  5^  for  collec- 
tion, what  must  be  the  assessment  to  pay  an  indebtedness  of 
$72675  ? 

27.  A  glazier  bought  of  Murphy  &  Diebold,  July  1,  1901, 
glass  to  the  amount  of  $324,  list  price,  from  which  he  re- 
ceived the  regular  discount  of  40,^.  He  got  an  additional 
discount  of  5^  for  cash.     Make  out  and  receipt  the  bill. 

28.  The  amount  at  6^  for  2  years,  5  months,  18  days  is 
$746.20.     What  is  the  interest  ? 


REVIEW  WORK.  335 

29.  Bought  8000  bushels  of  wheat  in  Chicago  at  $.75  a 
bushel,  and  shipped  it  to  New  York,  where  my  agent  sold  it 
at  $.87^-  a  bushel.  His  commission  was  2^,  and  other  ex- 
penses were  $325.     What  was  my  gain  ? 

30.  A  druggist  marked  goods  to  sell  at  40^  gain.  lie  lost 
lOfo  of  sales  in  bad  debts,  and  paid  10^  for  collecting.  What 
Was  his  net  gain  per  cent  ? 

31.  The  net  earnings  of  a  stock  company  are  $22425,  and 
the  capital  stock  is  $215000.  What  per  cent  dividend  can  be 
declared,  no  surplus  being  reserved  ? 

32.  In  selling  hay  for  $15  a  ton  I  lose  10^.  At  what  price 
must  I  sell  it  to  gain  10,'^^  ? 

33.  Wlien  green  hams,  bought  at  8  cents  a  pound,  waste 
10^  in  curing,  at  what  price  must  they  be  sold  to  gain  30ji^ 
on  the  cost  ? 

34.  A  rectangular  garden,  40  feet  long  and  20  feet  wide,  is 
enlarged  10^  in  each  dimension.  Find  the  per  cent  of  in- 
crease in  area. 

35.  The  proceeds  of  a  30-day  note,  discounted  at  6^,  are 
$143.28.     Find  the  face. 

36.  In  October  a  contractor  counted  the  cost  of  a  proposed 
building,  and  adding  10^  for  his  profit,  put  in  his  bid,  which 
was  accepted.  Before  he  began  work  in  April,  labor  and 
material  had  advanced  5^.  What  per  cent  profit  did  he 
realize  ? 

37.  A  note  for  $486,  dated  Sept.  7,  1895,  was  endorsed  as 
follows  :  Rec'd  March  22,  1896,  $125  ;  Nov.  29,  1896,  $150 ; 
May  13,  1897,  $120.  What  was  due  April  19,  1898,  the  rate 
of  interest  being  6fc  ? 

38.  A  merchant  buys  goods  on  6  months'  credit.  At  the 
end  of  one  month  he  borrows  money  at  6^  per  annum  and 
pays  the  bill,  getting  a  discount  of  5^.  How  much  does  he 
save  on  a  bill  of  $3650  ? 

39.  AVhat  is  the  difference  between  the  annual  and  the 
compound  interest  on  $300  for  2  years  at  6  per  cent  ? 


336  SCHOOL  ARITHMETIC. 

40.  A  and  B  each  had  152900.     A  invested  his  money  in 

7^  raih-oad  bonds  at  132^,  brokerage  ^.  B  loaned  liis  at  5^ 
simple  interest.  Which  received  the  larger  income,  and  how 
much  ? 

SUPPLEMENTARY  EXERCISES  (FOR  ADVANCED  CLASSES). 

601.  1.  The  difference  between  the  interest  on  $600  and 
that  on  $750  at  6fo  for  a  certain  time  is  $18.75.  What  is  the 
time  ? 

2.  Find  the  simple  interest  at  8^  on  $5000  belonging  to  a 
boy  12  yr.  6  mo.  15  da.  old,  and  remaining  on  interest  until 
he  is  of  age. 

3.  If  the  use  of  $3750  for  8  months  is  worth  $68.75,  what 
sum  is  that  whose  use  for  two  years  4  months  is  worth  $250  ? 

4.  A  owes  B  $1500  due  in  1  yr.  18  mo.;  he  pays  him  $300 
cash,  and  gives  a  six-month  note  for  the  balance.  What  is 
the  face  of  the  note,  money  being  worth  6  per  cent  ? 

5.  Bought  goods  at  25,  20,  15,  and  10^  off.  If  the  sum  of 
my  discounts  was  $162.30,  what  was  the  list  price  ? 

6.  I  sent  my  agent  $1364.76  to  be  invested  in  pork  at  $6  a 
barrel,  after  deducting  his  commission  of  2^.  How  many 
barrels  of  pork  did  be  buy  ? 

7.  I  sold  a  pig  at  a  loss  of  25^.  Had  it  cost  me  $1  more, 
my  loss  would  have  been  40^.     Find  its  cost. 

8.  A  merchant  bought  goods  for  $150,  and  sold  -^  of  them  at 
a  loss  of  4^.  What  per  cent  must  that  selling  price  be  in- 
creased so  that  by  selling  the  rest  at  the  increased  rate  he 
may  gain  4^  on  the  whole  transaction  ? 

9.  A  man  has  an  opportunity  of  investing  $10000  in  min- 
ing stock  at  125,  paying  8^  dividend ;  in  railroad  stock  at 
85,  paying  5^  ;  or  he  can  loan  his  money  at  6^.  Which  would 
yield  the  largest  income,  and  how  much  ? 

10.  If  I  sell  oats  at  42^^  a  bushel,  my  gain  is  only  f  of 
what  it  would  be  if  I  should  sell  at  56^^  a  bushel.  Find  the 
price  paid  for  them. 


PROPORTIONAL  PARTS.  337 

11.  A  farmer  paid  176  for  calves  and  sheep,  paying  $3  for 
calves  and  $2  for  sheep.  He  sold  ^  of  his  calves  and  f  of  his 
sheep  for  $23,  thereby  losing  8^  on  their  cost.  How  many 
of  each  did  he  buy  ? 

12.  The  cost  of  a  bridge  was  $1260.52,  which  was  raised 
by  a  tax  upon  the  property  of  the  town.  The  tax  levy  was 
d^  mills,  and  the  collector's  commission  was  3^^.  What  was 
the  valuation  of  the  property  ? 

13.  I  wish  to  borrow  $350  from  a  bank  for  90  days.  For 
what  sum  must  I  give  my  note,  if  the  banli  discounts  it  at 
6fo  ?     Write  the  note. 


PROPORTIONAL.    PARTS. 

502.  1.  A  and  B  bought   15  horses.      For  every  one  A 
bought,  B  bought  2.     How  many  did  each  buy  ? 

Queries,— How  many  did  both  buy  at  one  purchase  ?    How  many  such 
purchases  were  made  ?    Then  how  many  did  A  buy  ? 

2.  Divide  $25  between  May  and  Ned,  giving  Ned  $2  as 
often  as  you  give  May  $3. 

3.  A,  B,  and   C  ate  27  peaches.     While  A  ate  two,  B  ate 
three,  and  C  ate  four.     How  many  did  each  eat  ? 

Queries.— How  many  did  all  eat  while  A  ate  two  ?    Then  how  often 
did  A  eat  two  ? 

WRITTEN    EXERCISES. 

503.  1.  Divide  $132  between  two  persons  so  that  their 
share  shall  be  in  the  ratio  of  5  to  7. 

$5  +  $7  =  $12. 

Ski  32  -^  $12  r=r  11  ^^^  ^^^^  ^  ^^  often  as  the  other  gets 

11  X  $5  =  $55  ^^'     I'hese  sums  can  be  distributed  11 

11  X  $7  =:  $77*.  ^""^^• 

2.  Divide  $100  into  parts  proportional  to  ^  and  -J. 
■    Suggestion,     i  =  f,  and  i  =  |  ;  hence,  ^  :  ^  =  3 :  3. 


338  SCHOOL   ARITHMETIC. 

3.  The  sum  of  two  numbers  is  187.  What  are  the  num- 
bers if  they  are  to  each  other  as  6  to  11  ? 

4.  Divide  533  into  three  parts  which  shall  be  to  one 
another  as  2,  3,  and  5. 

5.  Three  numbers  are  to  one  another  as  3,  5,  and  9,  and 
one  half  their  sum  is  153.     What  are  the  numbers  ? 

6.  Divide  693  into  five  parts  which  shall  be  proportional  to 
12,  7,  6,  5,  and  3. 

7.  For  every  $15  earned  by  a  man,  his  three  sons  each 
earned  $9,  and  his  two  daughters  each  16.  If  all  earned 
$5616  in  a  year,  how  much  did  each  earn  in  a  week  ? 

PARTNERSHIP. 

504.  1.  A  and  B  engaged  in  business,  A  investing  $500 
and  B  $1000.  If  the  gain  was  $750,  what  was  each  one's 
share  ? 

Queries. — Should  A  and  B  share  the  gain  in  proportion  to  their  in- 
vestments ?  Then  $750  must  be  divided  into  parts  proportional  to  what  ? 
Since  A  furnishes  ^  of  the  capital,  what  part  of  the  gain  should  he 
receive  ?    Is  the  ratio  i  :  §  equal  to  the  ratio  $500  :  $1000  ? 

2.  D,  A,  and  E  formed  a  partnership,  D  putting  in  $1000, 
A  $2000,  and  E  $3000.  At  the  end  of  the  year  they  had 
gained  $1800.     What  was  each  one's  share  ? 

505.-  An  association  of  two  or  more  persons  for  the  purpose 
of  carrying  on  business  is  called  a  Partnership. 

The  persons  thus  associated  are  called  partners,  and  to- 
gether they  form  a  company  ov firm. 

506.  The  money  (or  its  equivalent)  invested  by  all  the 
partners  is  called  the  capital  of  a  firm.  Its  debts  are  called 
Uahilities. 

507.  Principle. — The  profits  and  losses  of  a  firm  are 
shared  in  proportion  to  the  parts  of  the  capital  invested  hy 
each  partner. 


PARTNERSHIP.  339 

WRITTEN    EXERCISES. 

508.  1.  A,  B,  and  0  formed  a  partnership,  A  furnishing 
$2000  of  the  capita],  B  $3000,  and  C  $7000.  What  was  each 
partner's  share  of  the  $4800  gained  ? 

(a) 
$2000  4-  $3000  +  $7000  =  $12000. 
$4800  -4-  $12000  =  .4,  or  f. 
i  of  $2000  =  $800,    A's  share. 
?  of  $8000  =  $1200,  B's     ♦' 
i  of  $7000  =  $2800.  C's     " 

(b) 
$2000  +  $3000  +  $7000  =  $12000. 
Gain  on  $1  of  capital  =  $4800  ^  12000,  or  $J. 
A's  gain  on  $2000  =  2000  x  $s,  or  $800. 

2.  R,  0,  and  H  are  partners.  R  put  into  the  business 
$4500,  0  16000,  and  II  $7500.  Tlieir  first  year's  gain  was 
$3750.     What  was  each  one's  share  ? 

3.  G,  11,  and  K  formed  a  partnership,  G  furnishing  $3750, 
H  $0250,  and  K  $8750.  When  tliey  dissolved,  H  received  $1350 
as  his  share  of  the  gain.     How  much  did  G  and  K  receive  ? 

4.  Smith  invested  $4200  in  a  business,  and  his  first  year's 
profit  was  $840.  His  partner,  Jones,  received  $1120  profit 
in  the  same  time.     How  much  did  Jones  invest  ? 

5.  B  and  0  formed  a  partnership  with  a  capital  of  $7500, 
and  realized  a  profit  of  $12000.  What  was  each  one's  share 
of  the  gain  if  B's  investment  was  |  of  C's  ? 

6.  R,  Q,  P,  and  Y,  partners,  have  a  capital  of  $300,000. 
At  the  end  of  a  year's  business  R  received  $12000  as  his  share 
of  the  profit,  Q  received  $10000,  P  $15000,  and  Y  $13000. 
How  much  did  each  invest  ? 

7.  T>,  L,  and  G,  who  were  engaged  in  business  three  years, 
made  an  annual  profit  of  $7200.  During  the  first  year  D 
owned  -I,  L  ^,  and  G  ^  of  the  stock  ;  during  the  second  year 
each  owned  ^  of  it ;  and  during  the  last  year  G  owned  ^  of 
the  stock,  while  the  other  half  was  equally  divided  between 
L  and  D.    What  was  each  partner's  share. of  the  total  profits? 


340  SCHOOL   ARITHMETIC. 

8.  A  and  B  bought  a  lot  for  11500,  agreeing  to  pay  1500 
cash  and  $1000  in  3  months.  A  pays  the  $500  cash,  and  B 
pays  the  $1000  three  months  later.  Nine  months  after  the 
purchase  they  sold  the  lot  for  $1750.  What  was  each  one's 
share  of  the  gain  ? 

The  use  of  $500    for  9  mo.  =  the  use  of  9  x  $500,  or  $4500,  for  1  mo. 
The  use  of  $1000  for  G  mo.  =  the  use  of  6  x  $1000,  or  $6000,  for  1  mo. 
Hence  the  respective  gains  are  proportional  to  4500  and  6000. 

9.  D,  E,  and  F  gain  in  trade  $8000.  D  furnishes  $12000 
for  6  mo.;  E,  $10000  for  8  mo.;  and  F,  $8000  for  11  mo. 
What  is  each  man's  share  of  the  gain  ? 

10.  X,  Y,  and  Z  hired  a  pasture  for  $420.  X  put  in  6 
horses  for  9  weeks,  Y  9  horses  for  8  weeks,  and  Z  12  horses 
for  7  weeks.     How  much  should  each  pay  ? 

11.  A,  B,  and  C  contribute  capital  to  a  business  as  follows: 
A  $3000  for  12  months,  B  $4000  for  10  months,  and  C  $5000 
for  8  months.  Their  profits  are  $900.  AYhat  is  the  gain  of 
each  ? 

12.  A,  B,  and  C  hired  a  pasture  for  $452.  A  put  in  12 
horses  for  15  weeks,  B  80  sheep  for  8  weeks,  and  C  18.cows  for 
20  weeks.  How  much  should  each  pay  if  a  cow  eats  as  much 
as  3  sheep,  and  5  sheep  eat  as  much  as  a  horse  and  2  sheep  ? 

13.  K  rented  a  house  for  $720  a  year.  After  3  months  B 
moved  in  with  him,  agreeing  to  pay  his  share  of  the  rent. 
Five  months  later  C  also  moved  in  on  the  same  conditions. 
How  much  of  the  $720  did  each  pay  ? 

14.  In  a  certain  company  B  has  3  times  as  much  capital  as 
A,  and  C  has  |  as  much  as  the  other  two.  What  is  each 
one's  share  of  a  profit  of  $393  ? 

15.  M  hired  a  rig  for  $10  to  drive  from  Salem  to  Manor,  a 
distance  of  10  miles,  and  back  again.  At  Derby,  midway 
between  the  two  places,  he  took  in  L,  who  agreed  to  pay  his 
proportional  share  of  the  expense  if  allowed  to  ride  to  Manor 
and  back  to  Derby.     How  much  should  L  pay  ? 


GENERAL   REVIEW  WORK. 

ORAL   EXERCISES. 

609.  1.  If  f  of  an  acre  of  land  costs  ^  of  $120,  what  will 
5  acres  cost  ? 

2.  What  will  3  ounces  of  silver  cost  if  half  a  pound  costs 
$4.20  ? 

3.  How  many  3-inch  squares  are  there  in  a  piece  of  paper 
2  yards  long  and  3  feet  wide  ? 

4.  At  two  cents  a  foot,  how  much  will  8  rods  of  wire  cost  ? 

5.  If  .3  of  John's  money  equals  f  of  Kate's,  and  both  have 
27  cents,  how  much  has  each  ? 

6.  How  many  cubic  inches  are  there  in  a  piece  of  scantling 
^  of  yard  long  and  a  foot  square  at  the  ends  ? 

7.  After  spending  4  of  his  money,  Henry  had  $2  less  than 
half  his  money  left.     How  much  had  he  at  first  ? 

8.  What  is  the  average  cost  of  25  cows,  if  13  of  them  are 
bought  at  $50  a  head,  and  the  others  at  $75  a  head  ? 

9.  If  4  of  one  number  is  135,  and  f  of  another  is  7^,  what 
is  the  sum  of  the  two  numbers  ? 

10.  What  is  the  number  whose  half  exceeds  its  third  by 
126? 

11.  Forty  per  cent  of  George's  marbles  equals  ^  of  Tom's, 
and  both  have  54.     How  many  has  each  ? 

12.  A  man  sold  50  acres  of  his  land,  and  had  37^^  of  it  left. 
How  many  acres  had  he  at  first  ? 

13.  In  a  mixture  of  grain  there  are  50  bushels  of  oats,  40 
of  corn,  and  15  of  wheat.    What  part  of  the  mixture  is  each? 

14.  A  is  5  miles  ahead  of  B,  and  walks  3^  miles  while  B 
walks  4.    How  many  miles  will  B  walk  before  overtaking  A  ? 


34:2  SCHOOL  AUITHMETIC. 

15.  Divide  42  apples  between  Edna  and  May  so  that  Edna 
will  have  -^  of  |^  as  many  as  May. 

16.  When  money  is  worth  5^,  what  must  I  pay  for  the  use 
of  130  for  3  years  8  months  ? 

17.  If  7  be  added  to  both  numerator  and  denominator  of 
the  fraction  |,  how  much  will  the  value  of  the  fraction  be 
increased  or  diminished  ? 

18.  When  money  was  worth  6^  a  year,  I  paid  $32  for  the 
use  of  1200.     How  long  did  I  have  it  ? 

19.  At  $1.50  a  cord,  what  is  it  worth  to  saw  a  cubical  pile 
of  wood  16  feet  long  ? 

20.  Ten  years  ago  A  was  half  as  old  as  B.  Ten  years 
hence  B  will  be  three  score  years  of  age.   How  old  is  each  now  ? 

21.  Ten  years  ago  Mr.  H  was  ^  as  old  as  he  will  be  30  years 
hence.     What  is  his  age  ? 

22.  Bought  shirts  at  115  a  dozen,  and  marked  each  to  be 
sold  at  a  profit  of  20,^^.     What  was  the  marked  price  ? 

23.  What  rate  of  income  do  I  receive  when  I  buy  6^  stocks 
at  50^  premium  ? 

24.  A  fruit  dealer  paid  15  cents  a  dozen  for  oranges,  and 
sold  them  at  the  rate  of  five  cents  for  two.  What  per  cent 
did  he  gain  ? 

25.  What  is  the  difference  between  .1  and  1^  ? 

26.  Sixty  per  cent  of  Nell's  money  is  75^  of  Ada's,  and 
both  have  $18.     How  much  has  each  ? 

27.  By  selling  eggs  at  4  cents  a  dozen  more  than  cost,  a 
grocer  made  25,^^.     At  what  price  did  he  sell  them  ? 

28.  A  man  gained  80  cents  on  a  bushel  of  berries  sold  at 
the  rate  of  25  cents  for  two  quarts.  What  was  the  cost  a 
quart  ? 

29.  Divide  $112  among  2  men  and  3  women,  giving  each 
man  twice  as  much  as  each  woman. 

30.  What  per  cent  of  a  yard  is  a  foot  and  three  inches  ? 

31.  What  is  the  least  sum  of  money  with  which  a  trader 
can  buy  sheep  at  $6  apiece  or  cows  at  $26  ? 


GENERAL   REVIEW   WORK.  3-|.3 

32.  IIow  many  tiles  each  G  inches  square  will  be  required 
to  cover  a  space  6  feet  square  ? 

33.  A  boy  sold  papers  at  the  rate  of  2^  for  5  cents,  50^  of 
which  is  profit.  How  many  papers  could  he  buy  for  a 
quarter  ? 

34.  IIow  many  square  yards  are  there  in  the  surface  of  two 
cubes  whose  edges  are  each  2  feet  6  inches  ? 

35.  A  room  is  ^  as  long  as  it  is  wide,  and  its  perimeter  is 
84  feet.     What  are  its  dimensions  ? 

36.  Sugar  worth  $1.55  is  weighed  in  a  false  balance  which 
gives  only  15Joz.  to  the  pound.  What  is  the  selling  price 
of  the  sugar  ? 

37.  What  per  cent  of  a  score  is  a  dozen  ?  What  per  cent 
of  a  dozen  is  a  score  ? 

38.  If  I  charge  $1.50  a  cord  for  sawing  wood  into  three 
pieces,  how  much  should  I  charge  for  sawing  it  into  five 
pieces  ? 

39.  One  square  rug  is  li  yd.  on  a  side,  and  another  is  1^ 
yd.  on  a  side.  If  the  larger  rug  costs  $1.44,  what  will  the 
smaller  cost  at  the  same  rate  a  square  yard  ? 

40.  A,  B,  and  C  bought  a  horse  for  $100,  A  paying  $20, 
B  $30,  and  C  $50.  They  sold  him  for  $175.  How  much 
did  each  gain  ? 

41.  F,  G,  and  H  have  $510.  If  G  has  |  as  much  as  F,  and 
H  has  f  as  much  as  F,  how  much  has  each  ? 

42.  If  2^  yards  of  cloth  make  a  pair  of  pants,  how  many 
pairs  can  be  made  from  a  piece  of  cloth  containing  40 
yards  ? 

43.  What  number  increased  by  6,  the  sum  multiplied  by  5, 
and  the  product  divided  by  10,  gives  3  as  a  product  ? 

44.  A  man  started  westward  from  London  and  traveled 
through  360°.  Did  his  watch  then  indicate  the  correct  time  ? 
Why  ? 

45.  How  many  sheep  are  worth  as  much  as  a  cow,  if  4 
cows  are  worth  one  horse,  and  2  horses  are  worth  48  sheep  ? 


844  SCHOOL  ARITHMETIC. 

46.  James,  who  lives  If  miles  from  the  schoolhouse,  goes 
to  school  5  days  each  week.  If  he  goes  home  for  lunch  every 
other  day,  in  how  many  days  does  he  walk  21  miles  ? 

47.  A  can  do  as  much  work  in  |  of  a  day  as  B  can  do  in  f 
of  a  day.  How  long  will  it  take  B  to  paint  a  house  that  A 
can  paint  in  18  days  ? 

48.  A  and  B  ran  a  mile,  A  beating  B  by  40  rods.  In  what 
time  can  B  run  a  mile,  if  A^s  time  in  the  race  was  7  minutes  ? 

49.  W  can  build  30  rods  of  fence  in  4  days,  and  R  can 
build  as  much  in  6  days  as  W  can  in  8  days.  In  what  time 
can  R  build  75  rods  of  fence  ? 

50.  A  man  who  had  3b  sheep  bought  three  times  as  many 
as  he  had,  and  then  sold  ^  of  all.     How  many  had  lie  left  ? 

51.  Harry  has  6a  cents,  which  is  f  as  many  as  Marie  has. 
How  many  cents  have  they  both  ? 

52.  How  many  square  feet  are  there  in  a  board  a  feet  long 
and  b  inches  wide  ? 

53.  The  dimensions  of  a  cube  are  a  inches.  How  many 
square  inches  are  there  in  5  of  its  sides  ? 

54.  In  a  school  there  'drep  pupils,  and  q  of  them  are  girls. 
How  many  boys  are  there  in  the  school  ? 

55.  If  a  yards  of  cloth  cost  b  dollars,  what  will  3  yards 
cost  ? 

56.  What  will  b  yards  of  cloth  cost  if  c  yards  cost  d  cents  ? 

57.  The  area  of  a  field  is  ab  square  rods,  and  the  length  is 
a  rods.     What  is  the  distance  around  the  field  ? 

58.  What  is  the  volume  of  a  cube  whose  edge  is  b  feet  ? 

59.  A  living-room  12  feet  square  and  10  feet  high  is  occu- 
pied by  5  persons.  How  many  cubic  feet  of  air  are  there  to 
each  person  ? 

60.  A  lot  two  rods  wide  is  planted  in  corn,  the  rows  being 
a  yard  apart.  How  many  rows  are  there,  no  row  being  nearer 
the  fence  than  1  foot  6  inches  ? 

61.  I  have  work  for  either  8  men  or  12  boys.  If  I  employ 
6  men,  to  how  many  boys  can  I  give  employment  ? 


GENERAL  REVIEW  WORK.  345 

62.  B  sold  a  bnggy  to  C,  gaining  20^,  and  C  sold  it  to  D 
at  a  loss  of  20fo.     If  D  paid  $150  for  it,  what  did  B  gain  ? 

63.  After  a  rain  it  was  found  that  there  was  half  an  inch 
of  water  in  a  box  3  feet  square.  What  was  the  volume  of 
the  water  in  the  box  ? 

64.  Said  A  to  B,  *'  I  have  as  many  quarters  as  you  have 
half-dollars,  and  we  together  have  $9."  How  much  had  each  ? 

65.  How  many  ounces  in  p  pounds  and  q  ounces  ? 

66.  What  is  the  quotient  of  a  fourths  -^  a  eighths  ? 

67.  There  are  946  pupils  in  a  school.  If  f  of  the  number 
of  girls  is  equal  to  f  of  the  number  of  boys,  how  many  of 
each  are  there  in  the  school  ? 

68.  What  per  cent  does  a  merchant  gain  on  his  investment 
if  20^  of  his  sales  is  profit  ? 

69.  I  sold  a  piece  of  land  so  that  ^  of  the  profit  equalled  J 
of  the  cost.     Find  the  gain  per  cent. 

70.  If  8  men  can  do  f  of  a  piece  of  work  in  9  days,  how 
many  men  can  do  the  whole  of  it  in  4  days  ? 

71.  How  many  hours  a  day  must  4  men  work  to  do  half  as 
much  work  in  10  days  as  15  men  can  do  in  4  days,  working 
10  hours  a  day  ? 

72.  A  can  hoe  -^^  of  a  row  of  corn  in  1  minute,  and  B  can 
hoe  2f  rows  in  an  hour.  In  what  time  can  they  together 
hoe  a  row  ? 

73.  Had  a  bin  contained  twice  as  much  oats,  and  the  oats 
been  used  one-fourth  as  fast,  the  oats  would  have  lasted  48 
weeks.     How  many  days  did  they  last  ? 

74.  Mv  pony  is  13  hands  high.     How  many  feet  is  that  ? 

75.  A  lady  gave  f  of  her  money  to  the  poor,  and  then 
found  f  as  much  as  she  had  given  away,  and  then  had  $30. 
How  much  had  she  at  first  ? 

WRITTEN     EXERCISES. 

510.  1.  A  wagon  box  is  8  ft.  long,  3  ft.  6  in.  wide,  and 
2  ft.  deep.     How  many  bushels  of  coal  will  it  hold  ? 


346  SCHOOL  ARITHMETIC. 

2.  A  and  B  can  do  a  piece  of  work  in  5  days,  B  and  C  in 
6  days,  and  A  and  C  in  10  days.  How  long  would  it  take 
each  alone  ? 

3.  A  saddle  cost  ^  as  much  as  a  horse,  and  the  horse  cost 
J  as  much  as  a  buggy.  If  all  cost  $500,  what  was  the  cost  of 
each  ? 

4.  What  number  is  that  from  which  if  '7-^  is  subtracted, 
f  of  the  remainder  is  25  ? 

5.  When  the  gold  dollar  was  worth  7^  more  than  the 
greenback  dollar,  how  much  gold  was  $371.29  in  greenbacks 
worth  ? 

6.  Sold  3  acres  of  land  for  $100  more  than  5  acres  cost, 
and  thus  gained  100^  on  -the  amount  sold.  What  was  the 
cost  an  acre  ? 

7.  After  losing  -f  of  his  money,  A  found  $15,  and  then 
lacked  -^j  of  having  his  original  amount.  How  much  did 
he  lose  ? 

8.  Divide  and  multiply  1  by  .001,  and  to  the  sum  of  the 
quotient  and  product  add  the  quotient  of  .01  -f-  50. 

9.  A  man  bought  5  shares  of  10^  bank  stock  ($100),  which 
yielded  him  Sfo.     What  did  it  cost  him  a  share  ? 

10.  A's  money  is  to  B's  as  7  :  11,  but  if  each  had  $9  more, 
A's  would  be  to  B's  as  5  :  7.     How  much  has  each  ? 

11.  A  ship  consumes  3^5-  of  its  coal  supply  each  day.  It 
starts  with  its  bunkers  f  full,  and  when  it  reaches  port  has 
only  -^~  of  its  supply  left.  How  many  days  were  occupied 
on  the  voyage  ? 

12.  Two  men  hired  a  pasture  for  $56.  A  puts  in  10  cows 
and  B  puts  in  36  horses.  If  a  cow  eats  twice  as  much  as  a 
horse,  how  much  should  each  pay  ? 

13.  Four  pipes,  each  2  inches  in  diameter,  empty  a  tank 
in  9  hours.  What  must  be  the  diameter  of  a  single  pipe  that 
will  empty  the  tank  in  the  same  time  ? 

14.  A,  B,  and  C  start  together  to  walk  around  a  race- 
track.    A  goes  once  around  in  2  hours,  B  in  3  hours,  and  C 


GENERAL  REVIEW  WORK.  347 

in  4  hours.     In  how  many  liours  will  all  he  together  again  at 
the  starting  point  ? 

16.  A's  money  added  to  ^  of  B's  equals  $2000.  How 
much  has  each,  if  A's  money  is  to  B's  as  3  to  4  ? 

16.  Bought  5^  stock  at  124f,  brokerage  i^.  What  rate  of 
interest  do  I  receive  on  my  investment  ? 

17.  If  24  sheep  are  worth  6  cows,  8  cows  worth  2  horses, 
and  3  horses  worth  90  pigs,  how  many  pigs  are  worth  a  dozen 
sheep  ? 

18.  Sold  a  lot  for  $G00,  payable  in  90  days  without  inter- 
est. Bought  it  back  the  same  day  for  $500,  payable  in  60 
days  without  interest.  How  much  did  I  gain,  money  being 
worth  8^  ? 

19.  The  assessed  valuation  of  a  town  is  $760,000.  What 
rate  must  be  levied  to  raise  $3610,  exclusive  of  the  collector's 
commission  of  5^  ? 

20.  A  offers  $300  cash  for  a  lot,  and  B  offers  $325,  payable 
in  9  months  without  interest.  Which  is  the  better  offer,  and 
how  much,  money  being  worth  6^  ? 

21.  Bought  a  horse  for  $150  on  a  year'*s  credit,  without 
interest,  and  sold  him  at  once  for  $150  cash.  How  much 
did  I  make,  money  being  worth  5^  ? 

22.  The  shadow  of  a  man  6  feet  tall  is  8  ft.  6  in.  long. 
Another  man's  shadow  is  7  ft.  9  in.  long.  How  tall  is  the 
latter  ? 

23.  What  is  the  least  quantity  of  milk  from  which  if  1 
quart  be  taken  the  remainder  can  be  exactly  measured  by 
either  a  2-quart,  a  4-quart,  a  6-quart,  or  an  8-quart  meas- 
ure ? 

24.  B  walked  twice  as  far  as  C  ;  but  if  he  had  walked  4 
miles  less,  and  C  6  miles  more,  he  would  have  walked  -J 
farther  than  C.     How  far  did  each  walk  ? 

25.  If  Tom  gives  May  a  penny,  each  will  have  the  same 
sum  ;  but  if  May  gives  Tom  a  dollar,  he  will  have  twice  as 
much  as  she  has  left.     How  much  has  each  ? 


348  SCHOOL  ARITHMETIC. 

26.  A  and  B  run  a  race,  their  rates  of  running  being  as 
17  :  18.  A  runs  2^  miles  in  16  minutes,  48  seconds.  B  the 
whole  distance  in  34  minutes.     What  is  the  distance  run  ? 

27.  A  owned  f  and  B  f  of  a  store,  but  they  took  C  into 
the  firm  and  reorganized  as  equal  partners.  If  C  paid  them 
14000,  what  was  A's  share  of  it  ? 

28.  If  16  yd.  of  cloth  cost  $56  when  wool  is  $.75  a  pound 
and  labor  $.20  an  hoar,  what  would  it  cost  when  wool  is  $.60 
a  pound  and  labor  $.25  an  hour,  if  24  lb.  of  wool  and  60  hr. 
of  labor  are  required  to  make  it  ? 

29.  Two  houses  cost  $8100,  and  f  of  the  cost  of  one  is 
equal  to  -^  of  the  cost  of  the  other.     What  is  the  cost  of  each  ? 

30.  A  sold  B  a  horse  for  ^  more  than  it  cost,  and  B  sold 
it  for  $80,  losing  I  of  its  cost.  How  much  did  A  pay  for 
the  horse  ? 

31.  I  paid  $214.20  for  a  piano  after  discounts  of  20^,  15^, 
and  10^  had  been  allowed.     What  was  the  list  price  ? 

32.  Schley  and  Sampson  were  partners  for  two  years, 
making  an  annual  profit  of  $5460.  During  the  first  year 
Sampson  owned  f  of  the  stock,  but  during  the  second  year 
Schley  owned  J  of  the  stock.  What  was  each  one's  share  of 
the  profit  ? 

33.  A  steamer  sails  a  mile  down  stream  in  five  minutes, 
and  a  mile  up  stream  in  7  minutes.  How  far  down  stream 
can  she  go  and  return  in  one  hour  ? 

34.  A  stable  30  ft.  long,  20  ft.  wide,  and  18  ft.  high  has 
two  gables  each  12  ft.  high.  Find  cost  of  painting  the 
outside  at  50^'  a  sq.  yd. 

35.  How  many  yards  of  carpeting  27  inches  wide  will 
cover  a  hall  45  ft.  long  and  32  ft.  wide,  the  strips  running 
lengthwise,  and  there  being  a  waste  of  ^  yard  in  matching 
the  pattern  ? 

36.  What  will  it  cost  to  plaster  the  walls  of  a  room  18|^  ft. 
long,  16^1  ft.  wide,  12  ft.  high,  at  11^  a  square  yard,  allow- 
ing nothing  for  openings  ? 


GENERAL  REVIEW  WORK.  34,9 

37.  How  many  board  feet  of  siding  5  in.  wide  will  be  re- 
quired to  cover  tlie  sides  of  a  house  40  ft.  long,  28  ft.  wide, 
20  ft.  high,  if  they  are  laid  4  inches  to  the  weather,  and  150 
sq.  ft.  are  deducted  for  doors  and  windows? 

38.  If  as  many  silver  dollars  as  possible  are  laid  on  the 
bottom  of  a  box  18  inches  long  by  12  inches  wide,  how  much 
space  will  be  left  uncovered  ? 

39.  The  fence  around  a  circular  field  is  1.19  miles  in 
length.     How  many  acres  inside  the  fence? 

40.  Home  is  20°  27'  14"  E.,  and  Washington  IT  3'  W. 
When  it  is  9  a.m.  at  Washington,  what  is  the  time  at 
Rome  ? 

41.  How  many  ounces  of  gold  in  a  IG-carat  chain  that 
weighs  3^  ounces  ? 

42.  At  one  point  an  eclipse  of  the  moon  was  seen  at  9 
A.M.,  at  another  point  at  11:30  a.m.  What  is  the  differ- 
ence in  the  longitude  of  the  two  places  ? 

43.  A  town  has  a  water  supply  of  104  gal.  a  day  for  every 
house.  If  the  number  of  houses  increases  ^,  and  the  total 
supply  diminishes  -j^,  what  will  be  the  daily  supply  to  a 
house  ? 


44.  Either  30  pears  and  20  apples  or  14  apples  and  42 
pears  will  just  till  a  basket.     How  many  of  either  will  fill  it  ? 

45.  A  and  B  receive  $1000  for  grading  a  street.  A  fur- 
nishes 3  teams  20  days,  B  5  teams  30  days.  If  A  receives 
1100  for  overseeing  the  work,  what  does  each  receive  of  the 
$1000  ? 

46.  A  ten-acre  field  was  divided  into  lots,  each  containing 
f  of  an  acre.  The  partial  lot  was  sold  at  the  rate  of  $300  an 
acre,  and  the  others  at  $150  each.  What  was  received  for  the 
field  ? 

47.  In  a  spelling  contest  there  were  75  words  given  ;  6 
contestants  spelled  74  words  each,  and  13  spelled  70  words 
each.     Find  the  average  per  cent  made  by  these  contestants. 


350  SCHOOL  ARITHMETIC. 

48.  City  lots,  200  feet  deep,  sell  for  180  a  front  foot.  What 
is  the  value  of  an  acre  at  that  rate  ? 

49.  A  train  runs  ^5  miles  an  hour.  How  far  can  I  ride  on 
it  and  walli:  back  at  the  rate  of  3|  miles  an  hour,  and  be  gone 
just  5  hours  ? 

60.  A  room  16  ft.  by  18  ft.  is  covered  with  carpet  27 
inches  wide,  and  the  smallest  possible  number  of  yards  of 
the  carpet  is  in  use.     How  many  yards? 

51.  Find  the  cost  of  a  bushel  of  ground  feed,  the  ingre- 
dients of  which  are  60  bushels  of  corn  at  45^,  90  bushels  of 
oats  at  32^,  and  26  bushels  of  rye  at  64^,  the  cost  of  grinding 
being  $6.20. 

52.  Two  men  together  received  $97.75,  but  one  received 
$18.25  more  than  the  other.     How  much  did  each  receive  ? 

53.  An  agent,  having  in  his  hands  $3150  of  his  principal's 
funds,  is  instructed  to  invest  it  in  barley  at  $.48  a  bushel, 
after  retaining  his  commission  of  5^.  How  many  bushels 
should  he  buy  ? 

54.  Two  districts  buy  a  road  machine  for  $285,  and  pay 
the  freight  from  the  factory,  one  district  paying  -|  and  the 
other  4^  of  the  entire  cost.  The  cost  of  the  first  district  be- 
ing $127.50,  how  mucli  was  charged  for  freight  ? 

55.  A  commission  merchant  sold  1014  bushels  of  oats  at  41 
cents  a  bushel,  paid  $33.74  freight  charges,  and' retained  3^^ 
commission.  How  much  should  he  remit  to  the  con- 
signor ? 

56.  The  Columbian  souvenir  half-dollar  weighs  192.9 
grains.  How  many  of  them  weigh  as  much  as  50  ordinary 
silver  dollars  ? 

57.  An  upright  pole  16  ft.  long  casts  a  shadow  5  ft.  4  in. 
long,  and  at  the  same  time  the  shadow  of  a  tree  is  found  to 
be  26  ft.  9  in.     How  high  is  the  tree  ? 

"58.  If  one  fifth  be  allowed  for  matching  and  waste,  how 
many  board  feet  of  inch  lumber  will  be  required  for  flooring 
and  ceiling  a  porch  17  ft.  4  in.  by  7  ft.  6  in.?  . 


aENERAL  REVIEW   WORK.  351 

59.  By  the  introduction  of  improved  machinery  in  a  cer- 
tain factory  it  was  found  that  7  men  could  do  the  work 
formerly  done  by  11  men.  What  per  cent  of  the  labor  re- 
quired to  turn  out  the  same  product  was  saved  by  using  the 
improved  machinery  ? 

60.  If  Tennessee  3^  bonds  are  selling  at  95,  how  much 
money  must  be  invested  in  them  to  secure  an  annual  income 
of  $750  ? 

61.  When  the  grade  on  a  road  is  1320  feet  to  the  mile, 
what  is  the  per  cent  of  grade  ? 

62.  If  the  interest  is  $19.07,  the  time  8  mo.  2  da.,  and  the 
rate  5^^^,  what  is  the  principal  ? 

63.  If  it  costs  $110  to  dig  a  cellar  40  ft.  long,  27  ft.  wide, 
and  4  ft.  deep,  how  much  will  it  cost  to  dig  a  cellar  36  ft. 
long,  30  ft.  wide,  and  5  ft.  deep  ? 

64.  The  running  time  of  a  train  from  New  York  to  Buf- 
falo is  8^  hours,  and  the  distance  is  440  miles.  If  stops  of 
5  minutes  each  are  made  at  Albany,  Utica,  Syracuse,  and 
Rochester,  what  is  the  average  speed  an  hour  ? 

65.  At  what  price  is  4J^  stock  equal  as  an  investment  to 
3^fo  stock  at  $87.50  a  share  ? 

66.  A  man  sells  22^  shares  of  5  per  cent  stock  at  105,  and 
buys  4  per  cent  stock  at  94|.  How  much  is  his  income 
diminished  ? 

67.  A  cooper  paid  $78.32  for  16488  barrel  staves.  Required 
the  price  per  M. 

68.  Bought  stock  at  8fo  below  par  and  sold  it  12^^  below 
par,  thereby  losing  $99.     How  much  did  I  invest  ? 

69.  Bought  4  loads  of  hay,  2750  lb.  each,  at  $20  a  ton,  and 
paid  for  it  with  a  GO-day  note  without  interest.  What  will 
be  the  proceeds  of  the  note  if  discounted  at  bank  im- 
mediately at  6^  ? 

70.  Railroad  stock  that  cost  $121.75  a  share  pays  a  semi- 
annual dividend  of  4^.  Required  the  rate  per  annum  of  in- 
come on  the  investment. 


352  SCHOOL  ARITHMETIC. 

71.  Divide  $744  among  A,  B,  and  C  so  that  f  of  A's  money 
will  equal  f  of  B's  or  |  of  C's. 

72.  The  sum  of  three  numbers  is  940.  The  first  equals  -f 
of  the  second,  and  the  second  equals  j^q-  of  the  third.  Find 
the  numbers. 

73.  Write  a  30-day  note  the  proceeds  of  which,  when  dis- 
counted at  a  New  York  bank  on  the  day  of  making,  shall  be 
$514. 

74.  Dewit  purchased  a  house  and  lot  for  $3300  ;  paid  $975 
for  repairs,  and  now  rents  the  premises  for  $30  a  month.  If 
he  expends  annually  for  taxes  $48.70,  and  for  incidental  re- 
pairs $35,  what  is  his  per  cent  of  annual  income  on  his  in- 
vestment ? 

75.  What  is  the  difference  between  the  true  and  the  bank 
discount  of  $200  for  60  days,  at  Gfo?     (No  grace.) 

76.  A  farm  is  worth  10^  less  than  a  store,  and  the  store 
20^  more  than  a  lot.  The  owner  of  the  lot  exchanges  it  for 
80^  of  the  farm,  thereby  losing  $850.  What  is  the  farm 
worth  ? 

77.  In  a  proportion  whose  ratio  is  12^,  the  first  number 
is  25  and  the  last  number  is  8.     What  is  the  third  number  ? 

78.  B  owns  a  square  mile  of  land,  and  D  owns  a  farm  of 
equal  area  whose  width  is  128  rods.  At  $1.25  a  rod,  how 
much  more  will  it  cost  to  fence  D's  land  than  B's  ? 

79.  A  farmer  agreed  to  give  his  hired  man  $100  and  two 
cows  for  a  year's  labor.  The  man  quit  work  at  the  end  of  10 
months,  receiving  the  cows  and  $70  as  a  fair  settlement.  At 
how  much  were  the  cows  valued  ? 

80.  When  a  railroad  company  is  declaring  quarterly  divi- 
dends of  1|^,  and  its  stock  is  quoted  at  125,  what  annual 
rate  of  income  does  an  owner  of  that  stock  receive  ? 

81.  A  young  man  puts  $10  in  a  savings  bank  each  month, 
making  his  first  deposit  Jan.  1,  1901.  How  much  will  there 
be  to  his  credit  Jan.  1, 1902,  if  the  bank  pays  4^  per  annum, 
and  adds  the  interest  at  the  end  of  each  quarter  ? 


GENERAL  REVIEW  WORK.  353 


SUPPLEMENTARY    EXERCISES  (FOR   ADVANCED    CLASSES). 

611.  1.  If  postage  stamps  are  ^  of  an  inch  long  and  f  of 
an  inch  wide,  how  many  will  be  required  to  cover  a  ceiling 
12  ft.  3  in.'  by  13  ft.  G  in.  ? 

2.  How  must  I  invest  in  3^  stock  at  90  so  as  to  get  the 
same  income  as  if  I  invested  $4950  in  the  same  stock  when  it 
is  quoted  at  99  ? 

3.  Divide  $14600  among  3  boys,  aged  9,  13,  and  17  years 
respectively,  so  that  if  invested  at  5^  simple  interest  each 
will  receive  the  same  amount  at  the  age  of  21. 

4.  A  owns  a  mine  worth  $11000,  which  pays  6^  on  his  in- 
vestment. Paying  $100  brokerage,  he  exchanges  the  mine 
for  bank  stock  at  109,  thus  increasing  his  annual  income 
$340,     What  dividend  does  the  bank  stock  pay  ? 

5.  A  pole  was  f  under  water.  The  water  rose  8  feet,  and 
then  there  was  as  much  of  the  pole  above  the  water  as  was 
previously  under  it.     Find  the  length  of  the  pole. 

6.  Two  equal  annual  payments  have  been  made  on  an  8^ 
note  for  $200,  dated  two  years  ago  to-day.  The  balance  due 
is  $44.     What  was  the  annual  payment  ? 

7.  A  cubic  foot  of  water  weighs  62.5  lb.,  and  lead  is  11.44 
times  as  heavy  as  water.  How  many  cubic  inches  are  there 
in  a  piece  of  lead  weighing  35  lb.  6  oz.  ? 

8.  M  and  K,  equal  partners,  found  on  settlement  that  M 
owed  the  firm  $240,  and  that  the  firm  owed  N  $260.  How 
much  should  M  have  given  N  to  square  the  account  ? 

9.  Snow  has  fallen  to  the  depth  of  25  cm.  If  12  cu  m. 
of  snow  produces  1  cu  m.  of  water,  find  the  volume  of  water 
produced  by  this  snow  on  one  acre  of  land. 

10.  A  rented  a  farm  from  B,  agreeing  to  give  B  |  of  all  the 
produce.  During  the  year  A  used  45  bu.  of  wheat,  and  at 
settlement  first  gave  B  18  bu.  to  balance  the  45  bu.,  and 
then  divided  the  remainder  as  if  neither  had  received  any. 
How  much  did  B  lose  ? 

23 


POWERS    AND    ROOTS. 
INVOLUTION. 

612.  1.  What  is  the  product  of  2  multiplied  by  itself,  or 
used  twice  as  a  factor  ? 

2.  How  often  must  3  be  used  as  a  factor  to  produce  9  ?  To 
produce  27  ? 

3.  What  is  the  product  of  a  used  twice  as  a  factor  ?  Used 
three  times  as  a  factor  ? 

513.  The  product  of  two  or  more  equal  factors  is  called  a 
Power.  The  product  of  two  equal  factors  is  called  the  second 
yoiver  ;  of  three  equal  factors  the  iJdrd power,  and  so  on. 

The  second  power  of  a  number  is  also  called  the  square  of  the  number, 
because  the  area  of  a  square  is  expressed  by  the  product  of  two  equal 
factors.     Why  is  the  third  power  of  a  number  also  called  its  cube  9 

1.  What  is  the  square  of  5  ?  Of  7  ?  Of  8  ?  Of  10  ?  Of 
i?     Off? 

2.  Find  the  third  power  of  2.  Of  4.  Of  7.  Of  \.  Of  |. 
Of  .3. 

514.  The  number  of  times  a  number  is  to  be  used  as  a 
factor  is  indicated  by  a  small  figure  placed  at  the  right  of 
the  number.     This  figure  is  called  an  Exponent. 

Thus,  a^  is  read  "a  square,"  or  *'a  to  the  second  power,"  and  means 
a  X  a;  d^  is  read  "a  cube,"  or  "a  to  the  third  power,"  and  means 
a  y.  a  X  a ;  a*  is  read  "a  to  the  fourth  power,"  and  so  on. 

Write  the  following  products  as  powers  : 

1.  3  X  3.  4.  5  X  5.  7.  9  X  9. 

2.  2  X  2  X  2.         5.  23  x  23.  8.  7  x  7  x  7. 

3.  «  X  «.  Q.  .a  X  a  X  a.         9.  bxbxbxbxb. 


POWERS  AND  ROOTS.  355 

Find  the  following  indicated  powers  : 

10.  8'  =  (   ).    ir  =  (   ).     ay  =  (   )    (HY  =  (   ). 
11. 5'  = '   ).      6'  =  (   ).     ay  =  (   )       .3'  =  (    ). 

12.  2-  =  ^     ).        3-  =.  ^    ).       (i)'  =  (     )  .3'  =  (     ). 

515.  Involution  is  the  process  of  finding  a  power  of  a 
number. 

WRITTEN     E/ERCISES. 

516.  1.  Find  the  third  power  of  12. 

2.  Find  the  square  of  10,  13,  15,  25,  36. 

3.  What  is  the  cube  of  8  ?    12  ?    20  ?    44  ? 

4.  Find  the  fourth  power  of  6,  10,  .5,  j. 

The  second  power  of  a  number  multipHed  by  the  second  power  equals 
the  fourth  power,  a'  x  a'  —  a*.  What  is  the  product  of  a'  x  a'  ? 
a*  X  a*  =.{    ). 

5.  Siiice  2"  X  2'  =  2*,  and  2*  x  2*  =  2",  what  is  the 
shortest  method  of  finding  the  IGth  power  of  2  ? 

6.  Find  tlie  third  power  of  |,  },  -f,  f,  .7. 

7.  What  is  the  square  of  1  ?     .1  ?     100  ?      .01  ?     2.5  ? 

8.  Can  a  number  ending  in  2,  3,  7,  or  8  be  a  perfect  square  ? 
Why  not  ? 

9.  How  many  figures  are  used  to  express  the  square  of  a  num- 
ber of  two  figures  ?     Tlie  square  of  a  number  of  one  figure  ? 

10.  What  is  the  difference  between  the  square  of  48  and 
the  cube  of  24  ? 

11.  How  much  does  the  cube  of  15  exceed  twice  its  square  ? 

12.  Which  is  greater  and  how  much — the  cube  of  ^  or  its 
square  ? 

Find  the  value  of  : 

13.  431  17.  6.25^  21.  (f)*.  25.  14="  -  U\ 

14.  46^  18.  .00b\  22.   (2|)^  26.  .500'. 

15.  14*.  19.  {{iY.  23.  5*  -  2\        27.  d9\ 

16.  3.75'.       20.  (if)'.  24.  (f)'  -  (f)^  28.  3'  +  2  x  5\ 

517.  Since  53  =  50  +  3,  the  square  of  53  maybe  obtained 
as  follows  : 


356 


SCHOOL  ARITHMETIC. 


(a) 
(50  +  3)  X     3  =  50  X  3  +  3 

(50  +  3)  X  50  =  50'  +  50  X  3 


X  3 


.-.  53'  =:  50'  +  'A  (50  X  3)  +  'i<' 
That   is,  53'  =  square  of  tens  +  twice    (tens  x  ones)  + 
square  of  ones. 

(b) 


53 
53 


53 
53 


159 
265 


9  rz:   3-^ 

150  =  50  X  3 
150  1=  50  X  3 
2500  =  50' 


2809  ■=  2809  =  50'  -h  2  (50  x  3)  +  3'. 

518.  Since  any  integral  num- 
ber expressed  by  two  or  more 
figures  may  be  regarded  as  com- 
posed of  tens  and  ones,  if  we 
represent  the  number  of  tens  by 
t  and  the  number  of  ones  by  o, 
we  have 

(t  +  of  =  f  +  Mo  +  o'.     Hence, 

The  square   of  a   numher  is 

equal  to  the  square  of  the  tens, 

^  plus   twice  the   tens  multiplied 

hy  the  ones,  plus    the  square   of  t  +  o 

the  ones. 

We  may  write  the  formula  thus: 
{t  +  o)'  =  f  +  {2t  +  o)  X  0.  Show 
why. 

Show  by  the  diagrams  that 

(a).  9'  =  (5  +  4)'  zrr  5'  +  4'  +  2 
X  (5'  X  4). 

(b).   (t  +  o)'  =  f  +  o'-i-  2to. 


5 4 


i' 

.  txo 

txo 

0^ 

POWERS  AND  ROOTS.  357 

Square  the  following  by  the  above  method  : 

1.  25.  4.  56.  7.  87.  10.  (4  +  3). 

2.  32.  6.  64.  8.  33.  11.  {a-^b). 

3.  43.  6.  71.  9.  98.  12.  (a  +  1). 

619.  The  cube  of  a  number  may  also  be  found  by  the 
above  method. 

1.  Raise  25  to  the  third  power. 

By  Art.  517,  25'  =  20'  +  2  (20  x  5)  +  5',  which  must  be 
multiplied  by  25,  or  20  +  5. 

25'  X  5    =  20'  X  5    +  2  (20  X  5')  +  5' 

25'  X  20  =  20'  +  2  (20'  x  5)  +  20  x  5' 

25'  =  20'  +  3  (20'  X  5)  +  3  (20  x  5')  +  5*. 
If  we  represent  the  number  of  tens  by  t  and  of  ones  by  o, 
we  have 

(t  +  oy  =  f+  3t'o  +  3(0'  +  o\     Hence, 

The  cube  of  a  ?iumber  is  equal  to  the  cube  of  the  tens,  plus 
three  times  the  product  of  the  square  of  the  tens  by  the  ones, 
plus  three  times  the  product  of  the  tens  by  the  square  of  the 
ones,  plus  the  cube  of  the  ones. 

We  may  write  the  above  formula  thus  : 

{t  +  oy  =  f  +  (3f  +  3to  +  o')  x  0.     Show  why. 
Cube  by  the  above  method  : 

2.  12.  5.  45.  8.  76.  11.  {a  +  b). 

3.  21.  6.  54.  9.  89.  12.  (2  +  3). 

4.  33.  7.  67.  10.  98.  13.  {a  +  1). 

EVOLUTION. 

520.  1.  What  is  one  of  the  two  equal  factors  of  16  ?  Of 
36  ?     Of  64  ? 

2.  What  is  one  of  the  three  equal  factors  of  8  ?  Of  27  ? 
Of  125  ? 


358  SCHOOL  ARITHMETIC. 

521.  One  of  the  equal  factors  of  a  number  is  called  a 
Root  of  tlie  number. 

The  square  root  is  one  of  the  two  equal  factors ;  the  cube 
root  one  of  the  three  equal  factors,  and  so  on. 

522.  The  process  of  finding  a  root  of  a  number  is  called 
Evolution. 

The  symbol  ^  is  called  the  Radical  Sig'ii,  and  calls  for 
the  square  root.  The  radical  sign  with  index  3  calls  for  the 
cube  root  ;  with  index  4  it  calls  for  one  of  the  four  equal 
factors,  and  so  on. 

Notes, — 1.  The  symbol  V  was  first  used  in  this  form  by  Rudolff  in 
1525. 

2.  The  square  root  is  also  indicated  by  the  fractional  exponent  I  ; 
the  cube  root  by  the  exponent  i,  etc. 

Find  the  root  called  for  : 

1.  V^5.        4.  ^27.        7.  Vi      lO.   A^lG.     13.  9*. 

2.  v/400.      5.  ^I.  8.  V^.     11.   ^^1.     14.  64*. 

3.  Vol.        6.  ^105.      9.  ^l\      12.   ^c\      15.  ^{a  +  x)\ 

SQUARE   EOOT. 

523.  1.  Since  1  =  1",  100  =  10^  10,000  =  100^  and  so  on, 
the  square  root  of  any  integral  number  between  1  and  100 
lies  between  what  two  numbers  ?  By  how  many  figures  is  it 
expressed  ? 

2.  The  square  root  of  any  integral  number  between  100 
and  10,000  lies  between  what  two  numbers  ?  By  how  many 
figures  is  the  root  expressed  ? 

524.  1.  The  square  root  of  any  integral  number  expressed 
by  one  or  two  figures  is  a  number  of  one  figure  ;  expressed  by 
three  or  four  figures  is  a  number  of  two  figures,  and  so  on. 

^.  If  an  integral  number  is  divided  into  groups  of  two 


POWERS  AND  ROOTS. 


359 


figures  each,  from  right  to  left,  the  number  of  groups  will  be 
equal  to  the  number  of  figures  in  the  root. 

Query. — May  the  left-hand  group  have  but  one  figure  ?    Why  ? 

525,  The  method  of  finding  the  square  root  of  numbers  is 
derived  from  the  identity, 

{t  +  oV  =  f  +  no  ^  d'^f  ^  {2i  +o)x  0.     (Art.  518.) 


WRITTEN     EXERCISES. 

526.  Find  the  square  root  of  1369. 
(a) 


(b) 


13  69 

t   +   0 

30  4-  7  =  37 

13  69  [37 

9  00  = 

(2t  +  o)x  0. 

9 

2t  =  2  x30  =  G0 

0=                     7 

4  69  = 
4  69 

67 

4  69 
4  69 

2t  +  0  =           07 

The  four  figures  of  the  number  show  that  the  root  is  expressed  by  two 
figures.  In  1300  the  greatest  tens-square  is  900,  and  its  square  root  is  30, 
which  is  the  tens  of  the  root. 

The  square  of  the  tens,  f,  is  subtracted,  and  the  remainder,  469,  con- 
tains twice  the  tens  x  the  ones  +  the  square  of  the  ones.  This  remainder 
is  largely  the  product  of  two  factors — twice  the  tens  x  the  ones,  of 
which  twice  the  tens,  or  60,  is  so  much  the  larger  that  it  maybe  used  as  a 
trial  divisor  ;  using  it  thus,  we  find  7  to  be  the  ones'  figure  of  the  root. 
Since  2to  +  o^  is  equal  to  (2^  +  o)  x  o,  the  seven  units  are  added  to  twice 
the  tens,  and  the  sum,  07,  is  multiplied  by  7. 

In  the  above,  2t,  or  60,  is  called  the  trial  divisor,  while  2t  +  o,  or  67,  is 
the  complete  divisor.  In  using  the  trial  divisor,  if  the  quotient  is  found 
to  be  too  great,  it  must  be  diminished. 

In  practice  the  operation  is  conveniently  performed  as  in  (b),  omitting 
unnecessary  ciphers.  The  first  group,  13,  contains  the  square  of  the 
tens'  number  of  the  root.  The  greatest  square  in  13  is  9,  and  its  square 
root  is  3.  The  square  of  the  tens  is  subtracted.  Twice  the  3  tens  is  6 
tens,  and  6  tens  is  contained  in  the  46  tens  of  the  remainder  7  times, 
giving  the  ones'  figure  of  the  root, 


360 


SCHOOL  ARITHMETIC. 


527.  We  may  illustrate  square  root  by  the  problem  of 
finding  the  side  of  the  square  whose  area  is  1369  square 
inches. 

The  square  root  of  1369  is  37.  The  square  of  37  =  (30  +  If  =:  30^  +  2 
(30  X  7)  +  T.  The  30  may  be  represented  by  a  square  ^  =  30  in.  on  a 
side.  The  2  (30  x  7)  may  be  represented  by  two  strips  each  ^  =  30  in. 
long  and  o  =  7  in.  wide,  while  the  7  may  be  represented  by  the  small 
square  o  =  7  in.  on  a  side  (Fig.  1). 

f  +  2to  +  0"  =  1369 
f  =    900 


t  =  50 


0=7 


6-30 


FiO.  2 


t=r30 

to=30\7 

Q* 

Fi$.l 


2^0  +  0  =  469 
That  is,  in  extract- 
ing the  square  root  of  1369,  the  large  square,  which  is 
^  =  30  in.  on  a  side,  is  first  removed,  and  a  surface  of 
469  sq.  in.  remains.  What  is  this  remainder  the  area  of  ? 
The  two  rectangles  and  the  small  square  have  one 
dimension,  o  =  7,  in  common.  If  placed  as  in  Fig.  2,  they  form  one  rect- 
angle whose  width  is  o  =  7,  and  whose  length  is  2^  +  o  =  60  -i-  7.  The 
area  is  therefore  expressed  by  (2/  -\-  o)  x  o,  or  (60  +  7)  x  7. 

In  finding  the  width  o,  we  are  obliged  to  use  2t,  or  60,  as  a  trial 
divisor,  since  the  whole  length  is  as  yet  unknown. 
469  -V-  2^  =  469  -r-  60  =  7. 
This  gives  (2^  +  o)  x  o  =  (60  +  7)  x  7  =  469. 
.-.   V^Vdi^  =  30  +  7  =  37,  the  number  of  inches  in  the  side. 

628.  We  may  apply  the  method  given  above  to  numbers 
of  more  than  two  groups  of  figures,  by  always  regarding  the 
part  of  the  root  already  found  as  so  many  tens  with  respect 
to  the  next  figure  of  the  root. 

1.  What  is  the  square  root  of  54756  ? 
(a) 

234 


•5  47  56 
4 


2  X  20  =:  40 

1  47 

40  +  'd  =    43 

1  29 

2  X  230  =  46 

0 

18  56 

460  +  4   =  46 

4 

18  56 

43 


(b) 

5  47  56 
4 

147 
1  29 

4 

18  56 
18  56 

234 


POWERS  AND  ROOTS.  361 

The  first  trial  and  complete  divisors  are  obtained  as  they  would  be  if 
the  given  number  were  547  ;  that  is  ^  =  20  and  o  =  3.  For  the  second 
divisors,  t  =  230  and  o  =  4. 

529.  Rule. — Beginning  at  ones,  separate  the  number  into 
groups  of  two  figures  each. 

Find  the  greatest  square  in  the  left-hand  group,  and  ivrite 
its  root  for  the  first  part  of  the  required  root. 

Subtract  the  square  of  this  root  from  the  left-hand  group, 
and  to  the  remainder  annex  the  next  group  for  a  dividend. 

Divide  this  dividend  by  twice  the  root  already  found,  con- 
sidered as  tens.  The  quotient  {or  the  qtiotient  diminished) 
will  be  the  next  figure  of  the  root. 

To  the  last  trial  divisor  add  the  part  of  the  root  last  found 
for  a  complete  divisor.  Multiply  this  complete  divisor  by  the 
part  of  the  root  last  found,  subtract  the  product  from  the 
dividend,  to  the  remainder  annex  the  next  group  for  a  new 
dividend,  and  proceed  as  before  until  all  of  the  groups  have 
been  thus  aimexed. 

1.  A  decimal  is  separated  into  groups  of  two  figures  each  by  begin- 
ning at  the  decimal  point,  and  its  root  is  found  precisely  as  the  root  of 
an  integer  is  found. 

2.  The  square  root  of  a  common  fraction  is  found  by  extracting  the 
square  root  of  numerator  and  denominator  separately,  or  by  reducing  it 
to  a  decimal  and  then  finding  the  root. 

3.  If  a  number  is  not  a  perfect  square,  ciphers  may  be  annexed,  and 
the  value  of  the  root  found  to  any  required  degree  of  approximation. 

Find  one  of  the  two  equal  factors  of  : 

1.  256.         4.  1024.         7.  4356. 

2.  441.         5.  2401.         8.  6080. 

3.  625.         6.  2809.         9.  7225. 

Find  the  square  root  of  the  following  : 

13.  1225.         17.  13225.         21.  143641. 

14.  1849.         18.  15625.         22.  173056. 

15.  4480.         19.  26001.         23.  499849. 

16.  9216.         20.  60516.         24.  801025. 


10. 

2025. 

11. 

.3249. 

12. 

.000144. 

25.  1234321. 

26.  5416.96. 

27.  97.8121. 

28.  31.4721. 

362  SCHOOL  ARITHMETIC. 

Extract  the  square  root  of  : 

29.  m  32.    m.  35.  6J. 

30.  ^\.  33.  T^iWr-  36.  3^. 

31.  im-  34.  ma.  37.  169AV- 

Find  the  value  to  four  decimal  places : 

38.  VU.        40.  V3.  42.   VX        44.  V^T 

39.  1/2".  41.  V20.         43.  V3r4.       45.   V52.321. 

APPLICATIONS   OF    SQUARE    ROOT. 

530.  1.  A  square  field  contains  40  acres.  Find  the  length 
of  a  side. 

2.  A  square  court  is  paved  Avith  3844  marble  slabs  8  inches 
square.     What  is  the  distance  around  the  court  ? 

3.  A  40-acre  field  is  three  times  as  long  as  wide.  Find  its 
length. 

4.  A  rectangular  field  is  60  rods  long  and  40  rods  wide. 
What  is  the  side  of  a  square  field  of  equal  area  ? 

6.  How  many  rods  of  fence  will  be  required  to  enclose  a 
square  farm  of  160  acres  ? 

6.  A  rectangular  farm  80  rods  wide  contains  160  acres. 
At  $1.45  a  rod,  how  much  will  it  cost  to  fence  it  ? 

7.  A  farmer  has  a  square  ten-acre  field  of  grass.  How 
many  times  will  he  have  to  mow  around  it  to  cut  the  grass, 
each  swath  being  5  feet  wide  ? 

8.  A  rectangular  field,  the  sides  of  which  are  in  the  ratio 
of  4  to  7,  contains  4032  sq.  rd.  Find  the  cost  of  fencing  it 
at  $2  a  rod. 

9.  What  is  the  difference  between  the  perimeters  of  two 
fields,  one  of  which  is  20.25  rd.  square,  and  the  other  20.25 
sq.  rd.  ? 

10.  An  army  of  7056  men  is  arranged  in  a  solid  square. 
How  many  men  in  rank  and  file  ?  How  many  soldiers  would 
be  required  to  make  another  row  around  the  square  ? 


POWERS  AND  ROOTS. 


363 


H 


531.  In  a  right-angled  triangle,  the  side  opposite  the 
right  angle  is  called  the  Hypotenuse. 

1.  Cut  from  a  cardboard  a  right  triangle  with  a  base  3 
inches  in  length  and  an  altitude  of  4  inches.  Find  by  actual 
measurement  tlie  length  of  the  hypotenuse. 

2.  Square  the  nunibers   representing  the  lengths  of  the 
three  sides,  and  find  whether  one  of  the 
squares  is  equal  to  the  sum  of  the  other 
two.     Which  one  ? 

3.  In  the  figure,  II  is  the  square  on 
the  hypotenuse,  B  the  square  on  the 
base,  and  A  the  square  on  the  altitude. 
How  many  small  squares  in  H  ?  How 
many  in  A  and  B  together  ? 

4.' Explain  by  the  figure  that  5'  =  3'  +  4'. 
+  B^ 

532.  The  square  on  the  hypotenuse  of  a  right-angled  tri- 
angle is  in  area  equal  to  the  sum  of  the  squares  on  the  other 
two  sides. 

Note. — This  is  known  in  geometry  as  the  Pythagorean  theorem,  be- 
cause it  is  supposed  to  have  been  first  proved  by  Pythagoras  (about  500  B.C.). 

533.  The  alcove  relation  is  expressed  in  the  equation  h^  = 
V'  +  «^  where  h  and  a  represent  the  base  and  altitude  re- 
spectively, and  h  the  hypotenuse. 

1.  Cut  a  cardboard  as  in  figure  1.  If  the  triangles  1,  2,  3,  4  are 
taken  away,  the  square  on  the  hypotenuse  of  a  right-angled  triangle 
remains  ;  and  if  the  two  rectangles  AP,  PB,  are  taken  away  from  the 
whole  figure,  the  sura  of  the  squares  on  the  two  sides  of  the  triangle  re- 
mains. Do  the  four  triangles  together  equal  the  two  rectangles  ?  Does 
this  prove  the  relation  Ih^  =  i^  +  a''  ? 


B 


Also,  H'  =  A^ 


B 


a+b 


Fig.l 


\ 

\: 

\ 

2 

h 


Fig.3 


a^ 

axb 

ap<b=ab 

b^ 

364  SCHOOL   ARITHMETIC. 

2.  Again,  cut  two  equal  squares  as  in  the  figures  2  and  3. 

From  Fig.  2,  (a  +  bf  =  h''  +  4:  (i  ah)  =  h^  +  2ab...{l) 

From  Fig.  3,  {a  +  hf  =  a"  +  h""  +  2ab (2) 

Show  from  (1)  and  (2)  that  Jt'  =  a""  +  b\ 

3.  If  the  sides  of  a  right-angled  triangle  are  6  ft.  and  8  ft.,  what  is  the 
length  of  the  hypotenuse  ? 

6  X  6  X  1  sq.  ft.  =  36  sq.  ft.,  the  square  on  one  side. 

8  X  8  X  1  sq.  f t.  =  64  sq.  ft. ,  the  square  on  the  other  side. 

36  sq.  ft.  4-  64  sq.  ft.  =  100  sq.  ft.,  the  square  on  the  hypotenuse. 

.'.  the  number  of  units  in  the  hypotenuse  =   V  100  =  10  ;  hence  the 


hypotenuse  is  10  ft.  long,     h  =  V  {b^  + 

a-)-- 

3  1/6^ 

+  8"^  =  y  100  =  10. 

Note. — Since  h^  =  P  +  a*,  we  get, 

by  subtra 
Hence  b 

Lcting  a^  from  both 

sides,  h'  -  a'  =  b\  or  &'  =  h^  -  a\ 

=   V  li"  -  a\     How 

find  the  value  oi  a  9 

Find  the  wanting  side  : 

Base.                           Altitude. 

Hypotenuse. 

1.        12                                   9 

(  ) 

2.        12                                16  • 

(  ) 

3.        15                             (     ) 

25 

4.      (     )                             28 

35 

5.  A  pole  100  feet  long  rests  against  the  top  of  a  wall  60 
feet  high.  How  wide  a  stream  could  flow  between  the  foot 
of  the  pole  and  the  wall  ? 

6.  Find  the  diagonal  of  a  room  18  feet  square. 

7.  IIoAV  far  will  a  man  walk  in  going  half  way  round  a 
square  farm  of  640  acres  ?     In  going  diagonally  across  it  ? 

8.  Find,  to  the  nearest  inch,  the  altitude  of  an  equilateral 
(equal-sided)  triangle  whose  side  is  6  inches. 

9.  While  a  ship  is  sailing  at  the  rate  of  12  miles  an  hour,  a 
sailor  walks  across  the  deck  at  the  rate  of  5  miles  an  hour. 
Find  his  rate  of  motion. 

10.  AVhat  is  the  grade  to  the  mile  on  a  road  on  which  the 
horizontal  distance  to  each  mile  is  5026  feet  ? 

11.  If  the  ridge  of  a  roof  is  9  feet  above  the  upper  floor  of 
a  house  24  feet  wide,  how  long  must  the  rafters  be,  allowing 
12  inches  for  cornice  ? 


POWERS  AND  ROOTS.  365 

12.  A  room  is  IG  feet  long,  12  feet  wide,  and  15  feet  high. 
Find  the  distance  from  any  lower  corner  to  the  opposite 
upper  corner. 

13.  Two  trees,  80  and  120  feet  high  respectively,  are  30 
yards  apart.     What  is  the  distance  between  their  tops  ? 

14.  Two  trains  leave  Richmond  at  the  same  time,  one  rnn- 
ning  north  at  the  rate  of  30  miles  an  hour,  the  other  east  at 
the  rate  of  40  miles  an  hour.  How  far  apart  will  tliey  be 
in  30  minutes  ? 

15.  Around  a  garden  80  feet  square  is  a  walk  containing 
one  sixth  as  much  area  as  the  garden  contains.  Find  the 
width  of  the  walk. 

16.  A  ladder  52  feet  long  stands  close  against  a  building. 
How  far  must  the  foot  be  drawn  out  that  the  top  may  be 
lowered  4  feet  ? 

17.  How  long  a  rope  will  wind  once  around  a  cylinder  10 
feet  long  and  6  feet  in  diameter,  commencing  at  one  end  and 
going  spirally  around  to  tlie  other  ? 

CUBE    ROOT. 

534.  1.  Since  1  =  1\  1000  =  10',  1,000,000  =  100',  and 
so  on,  the  cube  of  any  integral  number  between  1  and  1000 
lies  between  what  two  numbers  ?  By  how  many  figures  is  it 
expressed  ? 

2.  The  cube  root  of  any  integral  number  between  1000  and 
1,000,000  lies  between  wliat  two  numbers  ?  By  how  many 
figures  is  the  root  expressed  ? 

535.  1.  The  cube  root  of  any  integral  number  expressed 
by  one,  tivo,  or  three  figures  is  a  number  of  one  figure  ;  ex- 
pressed hy  four,  five,  or  six  figures  is  a  number  of  tiuo  figures, 
and  so  on. 

2.  If  an  integral  number  is  divided  into  groups  of  three 
figures  each,  from  riglit  to  left,  the  number  of  groups  will 
be  equal  to  the  number  of  figures  in  the  root. 


366 


SCHOOL  ARITHMETIC. 


Query. — May  the    left-hand  group  have  but  one  ov  two  figures  ? 
Why  ? 

536.  The  method  of  finding  the  cube  root  of  numbers  is 
derived  from  the  identity  (Art.  519)  : 
(t  +  oy  =  t'+  dt'o  +  3io'  +  o'=t'  -{-  (3r  +  3io  +  o')  x  o. 


WRITTEN    EXERCISES. 

537o  1.  Find  the  cube  root  of  46656. 
•  (a) 


46  656 


t  +  0. 
30  +  6  =  36. 


3r  =r  3  X  30'  =  2700 

19  656  : 

?>to  r^  3  X  30  X  6  ==    540 

o'  =  6'  =      36 

3^  +  dto  -h  o'  =  3276 

19  656 

27  000  =  r 


19  656  =  (Sr  +  3to   +  o')   X  o. 


(b) 


46  656  136 

27 


3   X  30'  =  2700 

3  X  30  X  6  =    540 

6'  =       36 

19  656 

3272 

19  656 

The  five  figures  of  the  number  show  that  the  root  is  expressed  by  tivo 
figures.  In  46000  the  greatest  tens-cube  is  27000,  and  its  cube  root  is  30, 
which  is  the  tens  of  the  root. 

The  cube  of  the  tens,  P,  is  subtracted,  and  the  remainder,  19656,  con- 
tains three  times  the  product  of  the  square  of  the  tens  by  the  ones  + 
three  times  the  product  of  the  tens  by  the  square  of  the  ones  +  the  cube 
of  the  ones.     Each  of  these  parts  contains  the  ones'  number  as  a  factor. 


POWERS  AND  ROOTS. 


36T 


Hence  the  19656  consists  of  two  factors,  one  of  which  is  the  ones*  num- 
ber of  the  root  ;  the  other  is  three  times  the  square  of  the  tens  +  three 
times  the  product  of  the  tens  by  the  ones  +  the  square  of  the  ones. 
Of  this  last  factor,  three  times  the  square  of  the  tens,  or  2700,  is  so 
much  the  larger  part  that  it  may  be  used  as  a  trial  divisor  ;  using  it 
thus,  we  find  7  to  be  probably  the  ones'  figure  of  the  root.  But  by  trial 
we  find  this  value  too  large,  and  we  must  take  one  less,  or  0,  for  the 
ones'  figure  of  the  root. 

Since  Zf^o  +  Zto^  +  o'  is  equal  to  (3^'  +  ^to  +  o')  x  o,  the  trial 
divisor  is  completed  by  adding  to  the  2700  the  3  x  (30  x  6)  =  540,  and 
e''  =  36;  the  sum,  3276,  is  multiplied  by  6. 

Notes. — 1.  In  finding  the  ones'  figure,  we  have  given  a  product  and 
the  greater  portion  of  one  factor  to  find  the  other  factor. 

2.  In  practice,  ciphers  are  omitted  for  convenience,  as  in  (b).  The 
first  group,  46,  contains  the  cube  of  the  tens'  number  of  the  root.  The 
greatest  cube  in  46  is  27,  and  the  cube  root  of  27  is  3.  Hence  3  is  the- 
tens'  figure  of  the  root.  We  then  divide  the  196  hundreds  of  the  re- 
mainder by  the  3  x  30^^  =  27  hundreds  to  get  the  ones'  number  of  the 
root. 

538.  We  may  illustrate  cube  root  by  the  problem  of  find- 
ing the  edge  of  a  cube  whose  volume  is  46656  cubic  inches. 


Here  t^  +  3^'o  +  3^0"  +  o'  =  46656,  whose  cube  root  is 


X  6')  +  6=" 


The  cube  of  36  =  (30  +  6)"  =  30'  +  3  (30'  x  6)  +  3  (30  x  6') 

The  30^  may  be  represented  by  a  cube  whose  edge  is  30  inches. 

The  3  (30'^  x  6)  may  be  represented  by  three  rectangular  solids,  each 
30  in.  long,  30  in.  wide,  and  6  in.  thick,  to  be  added  to  three  adjacent 
faces  of  Fig.  1. 

The  3  (30  x  Q"^)  may  be  represented  by  three  equal  rectangular  solids 
30  in.  long,  6  in.  wide,  and  6  in.  thick,  to  be  added  to  Fig.  2. 

The  6^  may  be  represented  by  the  small  cube  required  to  complete 
the  cube  of  Fig.  3. 


f  +  ^fo  +  Ma' 
t' 


+  0- 


=  46656 
=  27000 


^^ 


V 


ZL 


^ 


/ 


W'o  +  3^0"  +  o'  =  19656 
That  is,  in  extracting  the 
cube  root  of  46656,  the  large       Fig.  l.  Fig.  2.  Fig.  3.         Fig.  4. 

cube,  whose  edge  is  ^  =  30 

in.,   is  first  removed.      There  remain  19656  cu.  in.     Of  what  is  this 
the  volume  ? 


368 


SCHOOL  ARITHMETIC. 


The  seven  additions  to  the  cube  of  the  tens  have  one  dimension,  0  =  6, 
in  common.  If  these  seven  solids  were  laid  side  by  side  so  as  to  form 
one  solid,  the  area  of  its  base  would  be  3^^  +  Bfo  +  o"^,  or  3  x  30'^' -h  3 
(30  X  6)  +  C^  ;  its  height  would  be  0  =  6  ;  and  its  volume  would  be  the 
product  of  these  factors. 

In  finding  the  height  0,  we  are  obliged  to  use  3/^  or  2700  (trial 
divisor),  as  the  area  of  the  base,  since  the  whole  area  is  as  yet  unknown. 
If  this  gives  too  large  a  value  for  0,  we  must  take  one  less. 

19656  -f-  3f  =  19656  -4-  (3  x  30^^)  =  7  + . 

By  trial  we  find  this  too  large  ;  hence  we  must  take  0  =  6.  This 
gives 

{Sf  +  Sio  +  o'^)  X  0  =  (3  x  SO''  +  3  X  30  X  6  +  6')  X  6  =  19656. 

s 

•*•  V46656  r=  30  +  6  =  36,  the  number  of  inches  in  the  edge  of  the  cube. 

539.  The  methods  given  above  will  apply  to  numbers  of 
more  than  two  groups  of  figures  if  we  regard  the  pari  of  the 
root  already  found  as  so  many  tens  with  respect  to  the  next 
figure  of  the  root. 

1.  What  is  the  cube  root  of  1906G24  ? 


3  X 


1  906  624  1124 

1 

3  X  lO''  =r 

300 

906 

(10  X  2)  = 

60 

22  _- 

4 

364 

728 

3  X  120^  =: 

43200 

.   178  624 

120  X  4)  =. 

1440 

4'"  = 

16 

44656 

178  024 

540.  Rule. — Beginning  at  ones,  separate  the  number  into 
groups  of  three  figures  each. 

Find  the  greatest  cuhe  in  the  left-hand  group,  and  write  its 
root  for  the  first  part  of  the  required  root. 

Subtract  the  cube  of  this  root  from  the  left-hand  group,  and 
to  the  remainder  annex  the  next  group  for  a  dividend. 

Divide  this  dividend  by  three  times  the  square  of  the  root 


POWERS   AND  ROOTS.  369 

already  found,  considered  as  tens.  The  quotient  (or  the  quo- 
tient diminished)  will  be  the  next  figure  of  the  root. 

To  the  last  {trial)  divisor  add  three  times  the  product  of 
tlie  first  part  of  the  root,  considered  as  tens,  by  the  part  last 
found,  and  also  the  square  of  the  last  part,  for  a  complete 
divisor. 

Multiply  the  complete  divisor  by  the  part  of  the  root  last 
found,  subtract  the  product  from  the  dividend,  to  the  remain- 
der annex  the  next  group  for  a  netv  dividend,  and  proceed  as 
before,  uiitil  all  of  the  groups  have  been  thus  annexed. 

Notes. — 1.  A  decimal  is  separated  into  groups  of  three  figures  each 
by  beginning  at  the  decimal  point. 

2.  To  find  the  cube  root  of  a  common  fraction,  extract  the  cube  root  of 
numerator  and  denominator  separately  ;  or  change  to  a  decimal  and  then 
extract  the  root. 

3.  If  a  number  is  not  a  perfect  square,  ciphers  may  be  annexed, 
and  the  value  of  the  root  found  to  any  required  degree  of  approxima- 
tion. 

Find  the  cube  root  of  the  following  : 

1.  2744.  7.  74088. 

2.  4096.  8.  140608. 

3.  8000.  9.  226981. 

4.  24389.  10.  1860867. 

5.  10648.  11.  12167000. 

6.  42875.  12.  926859375. 

APPLICATIONS  OF  CUBE  ROOT. 

641.  1.  A  cubical  block  of  marble  contains  13824  cubic 
inches.     What  is  the  length  of  a  side  ? 

2.  A  cubical  cistern  holds  1000  gallons  of  water.  How 
deep  is  it  ? 

3.  How  many  square  inches  in  one  face  of  a  cube  of  granite 
whose  contents  are  5832  cubic  inches  ? 

4.  Find  the  length  of  the  diagonal  of  a  cube  whose  volume 
is  8000  cubic  inches. 


13. 

1%.  . 

14. 

mi. 

15. 

mh 

16. 

140.608. 

17. 

250.047. 

18. 

.970299. 

370  SCHOOL  ARITHMETIC. 

6.  A  cubical  bin  contains  500  bushels  of  wheat.  How 
deep  is  it  if  it  is  half  full  of  wheat  ? 

6.  A  square  box  16  in.  deep  will  hold  9.64276  bushels  of 
grain.     What  is  the  length  of  its  side  (inside  measure)  ? 

7.  A  miller  wishes  to  make  a  cubical  bin  that  will  hold  200 
bushels  of  grain.     What  must  be  its  depth  ? 

8.  What  is  the  length  of  the  edge  of  a  cubical  box  that  will 
hold  one  half  as  much  as  one  whose  edge  is  4  feet  ? 

9.  Two  thousand  gallons  just  fill  a  vat  whose  length  is 
twice  its  width,  and  whose  height  is  J  of  its  length.  Find 
the  length. 

10.  Assuming  that  cast  iron  weighs  7.15  times  as  much  as 
water,  find  the  edge  of  a  cube  of  such  iron  that  would  weigh 
a  ton. 

11.  The  width  and  depth  of  a  cistern  are  equal,  its  length 
is  twice  its  width,  and  it  will  hold  24891  gallons  of  water. 
What  are  its  dimensions  ? 

12.  By  cooling  a  red-hot  cube  of  iron,  the  length  of  each 
of  its  edges  was  diminished  by  6^.  Find  correct  to  three 
decimals  the  ratio  of  decrease  in  the  volume  of  the  cube. 


MISCELLANEOUS    PROBLEMS. 

ORAL    EXERCISES. 

542.  1.  The  sum  of  two  numbers  is  60,  and  their  differ- 
ence is  12.     What  are  the  numbers  ? 

2.  If  a  man  can  build  .3  of  a  fence  in  a  day,  in  what  time 
can  he  build  ^  of  it  ? 

3.  How  many  pies  will  be  needed  to  give  each  of  48  boys  f 
of  a  pie  ? 

4.  If  a  man  can  do  f  of  a  piece  of  work  in  a  day,  how  long 
will  it  take  2  men  to  do  .5  of  it  ? 

5.  If  a  bushel  and  3  pecks  of  potatoes  last  a  family  5  weeks, 
how  many  days  will  a  peck  and  a  half  last  them  ? 

6.  A  man  lost  12^^  by  selling  a  cow  for  $35.  What  per 
cent  would  he  have  gained  by  selling  her  for  150  ? 

7.  The  sum  of  two  numbers  is  45,  and  tlieir  difference  is 
25^  of  the  smaller  number.     What  are  the  numbers  ? 

8.  I  bought  eggs  at  20  cents  a  dozen.  Had  I  paid  a 
quarter  a  dozen  they  would  have  cost  a  dollar  more.  How 
many  did  I  buy  ? 

9.  A  man  spent  f  of  his  money  for  a  sleigh  and  ^  of  it 
for  a  book-case.     What  per  cent  of  his  money  had  he  left  ? 

10.  By  selling  books  at  a  profit  of  40^  an  agent  gained 
$250.     What  did  the  books  cost  him  ? 

11.  What  number  increased  by  its  half,  third,  and  fourth 
equals  50  ? 

12.  If  a  man  can  earn  a  dollar  in  f  of  a  day,  how  much 
can  he  earn  in  f  of  a  day  ? 

13.  What  number  diminished  by  12  is  equal  to  4  times 
-f^  of  the  number  ? 


372  SCHOOL  ARITHMETIC. 

14.  A  man  divided  his  farm  of  160  acres  between  his  two 
sons  in  the  ratio  of  ^  to  -J.     How  many  acres  did  each  receive  ? 

15.  One  half  of  the  money  in  my  purse  is  quarters,  ^  is 
nickels,  ^  is  dimes,  and  the  remainder  is  5  pennies.  What 
sura  is  in  my  purse  ? 

16.  B  and  C  together  have  $50,  and  B  has  a  dollar  more 
than  C.     How  much  has  each  ? 

17.  What  is  the  diagonal  of  a  table  whose  length  is  4  feet 
and  whose  width  is  75^  of  the  length  ? 

18.  I  borrowed  a  sum  of  money  for  3  years,  at  the  end  of 
which  time  I  repaid  the  loan  by  a  check  for  $22.50  more  than 
I  borrowed.  If  the  rate  M'as  5^,  what  was  the  sum  bor- 
rowed ? 

19.  A,  B,  and  C  shared  a  tract  of  land  in  the  ratio  of  1,  ^, 
and  .2.  C  received  4|-  acres  less  than  B.  What  was  the 
share  of  each  ? 

20.  Wliat  is  the  gain  per  cent  when  half  a  yard  of  cloth  is 
sold  for  what  f  of  a  yard  cost  ? 

21.  A  man  engaged  to  work  10  days  for  $30,  agreeing  to 
forfeit  $2  for  every  day  he  failed  to  work.  If  he  received 
$22.50,  how  many  days  was  he  idle  ? 

22.  A  lady,  being  asked  how  many  children  she  had,  re- 
plied, ''  If  I  had  twice  as  many  and  6  more  I  would  have  a 
dozen."     How  many  had  she  ? 

23.  A  fox  is  80  rods  ahead  of  a  hound,  and  runs  20  rods 
while  the  hound  runs  25.  How  far  will  the  hound  run 
before  he  catches  the  fox  ? 

24.  A  man  paid  $45  for  some  sheep.  Three  of  them  died 
and  I  of  the  remainder  were  sold  for  cost,  which  was  $30. 
How  many  were  sold  ? 

25.  Either  8  turkeys  or  12  ducks  are  needed  for  a  dinner. 
If  only  3  ducks  can  be  had,  how  many  turkeys  must  be  taken  ? 

26.  A  starts  on  a  journey  and  travels  27  miles  a  day  ;  7 
days  later  B  starts,  and  travels  36  miles  a  day.  In  how  many 
days  will  B  overtake  A  ? 


MISCELLANEOUS   PROBLEMS.  373 

27.  A  farmer  raised  60  bushels  of  potatoes,  and  the  crop 
was  1500^  of  the  seed.     How  many  bushels  were  planted  ? 

28.  A  man  sold  f  of  his  sheep  to  A,  -^  of  the  remainder  to 
B,  and  then  had  33  left.     How  many  did  he  sell  to  B? 

29.  If  f  of  A's  age  is  f  of  B^s,  and  the  sum  of  their  ages  is 
51  years,  what  is  the  age  of  each  ? 

30.  Three  men  hired  a  pasture  for  $114.  A  put  in  3  horses 
16  weeks ;  B,  5  horses  12  weeks ;  C,  8  horses  15  weeks.  How 
much  should  A  pay  ? 

31.  If  an  orange  and  a  half  cost  a  cent  and  a  half,  how 
many  oranges  may  be  bought  for  a  dime  ? 

32.  I  sold  a  watch  to  A  for  -^  more  than  it  cost  me;  he 
sold  it  for  $18,  thereby  losing  §  of  what  it  cost  him.  How 
much  did  I  pay  for  it  ? 

33.  A  is  f  as  old  as  B,  but  if  he  were  6  years  older,  he 
would  be  .  9  as  old  as  B.     How  old  is  each  ? 

34.  D's  money  is  $3  more  than  |  of  B's,  and  $5  less  than 
f  of  B's.     How  much  has  each  ? 

35.  If  3  boys  do  a  piece  of  work  in  9  hours,  how  long  will 
it  take  a  man  who  works  4^  times  as  fast  as  a  boy  ? 

36.  If  6  men  can  dig  a  ditch  in  3^  days,  how  much  time 
will  be  saved  by  employing  2  more  men  ? 

37.  A  boat  goes  ip  miles  an  hour  up  stream,  and  15  miles 
an  hour  down  stream.  How  far  can  she  go  and  return  in  10 
hours  ? 

38.  A  party  of  6  hired  a  coach.  If  there  had  been  2  more, 
the  expense  would  have  been  $1  less  for  each  person.  How 
much  was  paid  for  the  coach  ? 

39.  A  can  do  a  piece  of  work  in  12  days,  or  in  8  days  with 
B's  assistance.  After  they  work  together  6  days,  B  finishes 
the  work,  for  which  he  receives  $10.  How  much  should  A 
receive  ? 

40.  AVhat  time  after  midnight  are  the  hour  and  minute 
hands  of  a  clock  first  together  ? 


374  SCHOOL   ARITHMETIC. 


WRITTEN     EXERCISES. 

543.  1.  What  is  the  value  of  a  lot  .625  of  which  is  worth 
$1250  ? 

2.  Of  the  people  in  a  building  f  are  boys,  .375  are  girls, 
and  the  remainder,  which  is  22,  are  men.  How  many  girls 
in  the  building  ? 

3.  Tiie  quotient  is  3,  the  remainder  -fj,  and  the  divisor  3'\. 
What  is  the  dividend  ? 

4.  How  often  can  .125  be  subtracted  from  the  sum  of  125 
tenths  and  125  hundredths  ? 

6.  If  -|  of  an  acre  of  land  costs  $41f,  what  will  4|  acres 
cost  ? 

6.  Find  V^  x  ^/'d  correct  to  two  decimal  places. 

7.  The  minuend  is  .875,  and  the  remainder  is  one  less  than 
1500  thousandths.     What  is  the  subtrahend  ? 

8.  Explain  the  short  process  of  dividing  by  33^  ;  by  125, 

9.  What  will  be  the  cost  of  6160  lb.  of  coal,  at  $5.50  a 
ton  ? 

10.  Divide,  by  using  factors,  2875  by  48,  and  explain  the 
process. 

11.  Between  the  lightning  and  the  thunder  I  noted  12f 
seconds.  How  far  away  was  the  thunder,  if  sound  traveled 
1140  feet  a  second  ? 

12.  What  number  divided  by  either  3,  4,  8,  9,  12,  18,  24, 
or  36  leaves  a  remainder  of  3  ? 

13.  Making  no  allowance  for  mortar,  how  many  bricks 
8  in.  long  and  4  in.  wide  will  be  required  to  pave  a  walk 
40  yards  in  length  and  5  feet  in  width  ? 

14.  If  the  velocity  of  electricity  is  288,000  mi.  a  second, 
how  long  would  it  take  electricity  to  travel  around  the  earth, 
considering  the  circumference  to  be  24,900  mi.? 

15.  A  and  B  hirea  pasture.  A  puts  in  21  cows  and  B  puts 
in  35.     If  B's  part  of  the  rent  is  $185,  how  much  is  A^s  ?' 

16.  If  40  pupils  use  6  boxes  of  crayons,  200  in  a  box,  in 


MISCELLANEOUS  PROBLEMS.  375 

3  mo.,  how  many  boxes,  150  in  a  box,  will  be  required,  at 
the  same  rate,  to  supply  75  pupils  for  2  mo.  ? 

17.  How  many  books  the  size  of  this  would  be  required  to 
cover  the  floor  of  your  schoolroom  ? 

18.  How  many  cubic  feet  of  ice  will  an  ice-house  hold 
whose  dimensions  are  50  feet  by  30  feet,  and  18  ft.  high, 
allowing  2  ft.  above  and  below  and  on  each  side  for  saw- 
dust? 

19.  What  is  the  value  of  7^  +  .3  +  18  4- 4.5 -2^  x  If  ? 

20.  A  lot  40  ft.  by  120  ft.  is  enclosed  by  a  wire  fence  3 
wires  high.  If  25  feet  of  wire  weighs  a  pound,  and  a  pound 
costs  5  cents,  what  did  the  wire  for  the  fence  cost  ? 

21.  A  party  of  60  hired  a  boat.  Had  there  been  20  more, 
the  expense  of  each  would  have  been  reduced  i^l.  How  much 
was  paid  for  the  boat  ? 

22.  What  is  the  area  of  a  square  field  whose  perimeter  is 
160  rods  ? 

23.  The  perimeter  of  a  rectangular  field  is  240  rods,  and 
the  width  is  |-  the  length.     How  many  acres  in  the  field  ? 

24.  One  half  the  diagonal  of  a  rectangular  field  is  25  rods, 
and  the  width  is  30  rods.     What  is  the  area  of  the  field  ? 

25.  What  is  the  volume  of  a  rectangular  solid  a  feet  square 
and  b  feet  long  ? 

26.  One  evening  a  tree  45  feet  high  cast  a  shadow  75  feet 
long.  At  the  same  time  a  shadow  of  another  tree  was  160 
feet.     How  high  was  the  other  tree  ? 

27.  If  10  ounces  of  cotton  make  6'f  yards  of  cloth  a  yard 
wide,  how  much  will  be  required  to  make  12  yards  48  inches 
wide  ? 

28.  The  rafters  of  a  barn  are  25  feet  long,  and  their  ends 
are  40  feet  apart.     What  is  the  height  of  the  gable  ? 

29.  If  29  cows  average  9  quarts  of  milk  each  per  day 
throughout  the  year,  and  the  milk  is  sold  at  an  average  of 
7  cents  a  quart,  what  is  the  total  annual  profit  if  the 
expenses  are  178  a  head  ? 


376  SCHOOL  ARITHMETIC. 

30.  How  many  feet  of  lumber  in  12  planks,  each  18  feet 
long,  10  in.  wide,  and  3  in.  thick  ? 

31.  The  extreme  end  of  the  minute  hand  of  a  town  clock 
moves  19  inches  in  12  minutes.  What  is  the  length  of  the 
minute  hand  ? 

32.  Each  board  in  a  floor  56  feet  long  and  28  feet  wide  is 
14  feet  long  and  6  inches  wide,  and  it  is  held  in  place  by 
8  nails.  If  68  nails  weigh  a  pound,  what  is  the  weight  of 
the  nails  in  the  floor  ? 

33.  If  36  yards  of  cloth  cost  $54  when  wool  is  25{Z5  a  pound, 
what  will  25  yards  cost  when  wool  is  20^  a  pound  ? 

34.  A  carpenter,  a  mason,  and  a  painter  built  a  house  by 
contract  for  $3000.  The  carpenter  worked  108  days,  the 
mason  72  days,  and  the  painter  45  days.  The  material  used 
cost  $1775.  How  much  did  each  man  earn  if  carpenter's 
wages  were  $3  a  day,  mason's  $4,  and  painter's  $2.40  ? 

35.  A  load  of  120  bushels  consists  of  corn  and  rye  in  the 
ratio  of  7  bu.  of  corn  to  3  bu.  of  rye.  How  much  rye  must 
be  taken  away  that  the  corn  may  be  to  the  rye  as  10  is 
to  4? 

36.  A  garden  40  yards  square  is  surrounded  by  a  walk  24 
feet  wide.     What  part  of  an  acre  does  the  walk  contain  ? 

37.  What  is  the  diameter  of  a  circular  field  containing  10 
acres  ? 

38.  An  orchard  contains  7200  square  rods,  and  its  length 
is  to  its  breadth  as  f  to  f .     What  is  its  length  ? 

39.  A,  B,  and  C  can  build  a  fence  in  10  days.  A  can 
build  twice  as  much  as  B,  and  C  f  as  much  as  A.  In  what 
time  can  each  alone  build  the  fence  ? 

40.  The  property  of  an  insolvent  debtor  amounts  to  $3560, 
and  his  liabilities  to  $8900.  How  many  cents  on  the  dollar 
will  his  creditors  receive  ? 

41.  What  is  the  distance  from  a  lower  corner  to  the 
opposite  upper  corner  of  a  room  16  feet  long,  12  feet  wide, 
and  10  feet  high  ? 


MISCELLANEOUS  PROBLEMS.  377 

42.  The  entire  surface  of  a  cube  is  1014  square  inches. 
How  many  cubic  inches  does  it  contain  ? 

43.  Required  the  cost  of  fencing  a  square  40-acre  field  at 
$1.50  a  rod. 

44.  Measure  your  schoolroom  and  calculate  the  distance 
from  tlie  center  of  the  ceiling  to  a  lower  corner. 

45.  If  steel  rails  weigli  180  lb.  to  the  yard  and  cost  $12  a 
ton,  what  will  be  the  cost  of  rails  for  2  miles  of  railroad,  one 
of  which  has  double  track  ? 

46.  A  cow  is  tied  by  a  rope  20  feet  long.  Upon  how  many 
square  yards  can  she  graze  ? 

47.  An  iron  slab  is  4  feet  long,  3  feet  wide,  and  an  inch 
thick.  If  drawn  out  until  only  ^  of  an  inch  thick,  how  many 
square  feet  will  its  sides  contain  ? 

48.  A  square  park  is  surrounded  by  a  walk  2  rods  wide. 
The  area  of  the  walk  is  2  acres.  What  is  the  area  of  the 
square  ? 

49.  Find  the  value  of  the  following  lumber  at  $24  per  M  : 

Six  6  X  9  sills,  18  ft.  long. 
Thirty-two  2x8  joists,  18  ft.  long. 
Thirty-four  2x6  rafters,  22  ft.  long. 

50.  Bought  apples  at  $3  a  barrel.  Half  of  them  rotted. 
At  what  price  must  I  sell  the  remainder  in  order  to  gain 
33^^  on  the  amount  bought  ? 

51.  If  a  man,  starting  at  noon  from  Pittsburg,  could  travel 
at  the  rate  of  15  degrees  an  hour,  where  would  he  be  in  24 
hours  ?     Would  his  watch  be  too  fast  ? 

52.  How  many  revolutions  will  a  wheel  whose  radius  is  2 
ft.  3  in.  make  in  rolling  a  mile  and  a  half  ? 

53.  The  troy  pound  is  what  per  cent  of  the  avoirdupois 
pound  ? 

54.  The  base  of  an  isosceles  (two*  sides  equal)  triangle  is 
32  feet,  and  the  altitude  is  12  feet.  What  is  the  perimeter  ? 
Draw  the  figure. 


378  SCHOOL   ARITHMETIC. 

55.  A  vessel  can  sail  25  miles  an  hour  with  the  current, 
and  15  miles  against  it.  What  is  her  rate  of  sailing  in  still 
water  ? 

66.  A  ship,  whose  rate  of  sailing  in  still  water  is  10  miles 
an  hour,  sails  40  miles  up  a  stream  in  5  hours.  What  is  the 
rate  of  the  current  ? 

67.  Last  year  my  rent  was  $350,  which  is  12|^  less  than  it 
is  this  year.  How  much  has  my  rent  been  increased  this 
year  ? 

68.  If  the  water  is  25^  of  the  milk,  how  many  gallons  of 
each  in  a  mixture  containing  62^  gallons  ? 

59.  In  a  company  of  120  persons,  the  children  are  33-^^  of 
the  men,  who  are  75^  of  the  women.  How  many  children 
are  there  ? 

60.  Each  of  two  dealers,  A  and  B,  wishes  to  sell  an  organ 
to  C.  A  ask-ed  a  certain  price,  and  B  33^^  more.  A.  then 
reduced  his  price  25^,  and  B  his  price  30^,  and  C  took  both 
organs,  paying  $302.     What  was  B's  price  ? 

61.  A  druggist  pays  $3.50  a  pound  avoirdupois  for  3  lb.  of 
a  certain  article.  If  he  sells  it  at  the  rate  of  60  cents  a  troy 
ounce,  how  much  does  he  gain  ? 

62.  Find  the  cost,  at  $24  per  M,  of  225  2-inch  planks  18 
ft.  long  and  8  inches  wide. 

63.  The  rainfall  one  day  was  .08  in.  How  many  cubic 
feet  of  water  fell  on  an  acre  ? 

64.  The  cost  of  polishing  5  sides  of  a  cubical  block  of  mar- 
ble, at  20  cents  a  square  foot,  was  $144.  What  are  its  solid 
contents  ? 

65.  A  man  sold  two  lots  for  $3600  each.  On  one  he  gained 
33^^  and  on  the  other  he  lost  33^^.  Did  he  gain  or  lose, 
and  how  much  ? 

66.  In  a  mixture  of  silver  and  copper  weighing  55  ounces, 
there  are  15  ounces  of  silver.  How  much  copper  must  there 
be  added  that  tliere  may  be  If  ounces  of  silver  in  10  ounces 
of  the  mixture  ? 


MISCELLANEOUS  PROBLEMS.  379 

67.  Forrest  and  John  receive  the  same  salary.  Forrest 
saves  ^  of  his,  but  Jolin  spends  $200  a  year  more  than  For- 
rest, and  at  the  end  of  three  years  finds  himself  $150  in  debt. 
Required  tlie  salary  of  each. 

68.  A  2-inch  plank  is  16  feet  long,  18  inches  wide  at  one 
end,  and  tapers  regularly  to  a  point.  How  many  feet  board 
measure  does  it  contain  ? 

69.  How  long  will  it  take  a  men  to  do  a  piece  of  work  that 
b  men  can  do  in  c  days  ? 

70.  A  man  receives  a  salary  of  $1250  a  year,  and  3^^  of  his 
salary  equals  16f^  of  his  savings.  What  sum  does  he  save 
per  annum  ? 

71.  How  shall  I  mark  goods  which  cost  me  $240  that  I 
may  fall  14f^  and  still  gain  20  per  cent  ? 

72.  From  an  excavation  whose  length,  width,  and  height 
are  equal,  2197  cu.  ft.  of  earth  were  taken.  How  many  feet 
of  boards  will  be  required  to  cover  the  sides  and  bottom  of 
the  cavity  ? 

73.  A  family  consumes  47^  lb.  of  meat  a  week.  If  each 
member  of  the  family  consumes  J  lb.  daily,  how  many  in  the 
family  ? 

74.  A  grocer  sold  18  dozen  and  9  boxes  of  matches  from 
2  gross.     What  part  of  the  whole  did  he  sell  ? 

75.  How  many  sheets  of  tin,  each  18  x  30  in.,  will  it  take 
to  cover  a  roof,  each  side  being  24  ft.  long  and  16  ft.  3  in. 
wide  ? 

76.  A  commission  merchant  after  paying  $1605  for  various 
expenses  found  that  he  had  cleared  $2494.  What  amount 
did  he  collect  if.  his  rate  of  commission  was  5  per  cent  ? 

77.  A,  B,  and  0  invested  $10,800  in  business,  B  paying  f 
as  much  as  A,  and  C  -^^  as  much  as  A  and  B  together.  The 
profit  the  first  year  was  33^^  of  the  capital.  How  much 
should  each  •receive  ? 

78.  Three  men  rent  a  room  for  one  year,  four  months,  at 
the  rate  of  $210  a  year.     The  first  man  paid  $122.50  for  the 


380  SCHOOL  ARITHMETIC. 

time  he  occupied  the  room  ;  the  second  man  occupied  the 
room  for  4  mo.,  and  the  third  for  the  remaining  time.  If 
each  paid  according  to  the  time  he  occupied  the  room,  how 
much  should  the  last  two  pay  ? 

79.  A  man  drew  out  of  the  bank  \  of  his  money  and  $16 
more  ;  then  ^  of  the  remainder  less  $20.  He  then  had  $182 
remaining  in  the  bank.     How  much  had  he  at  first  ? 

80.  How  many  times  will  a  wagon  wheel  12  ft.  6  in.  in  cir- 
cumference turn  round  in  going  10  mi.  24  rd.  4  ft.  ? 

81.  In  a  pile  of  wood  16  feet  long  and  16  ft.  wide,  there 
are  12  cords.     How  high  is  it  ? 

82.  How  many  times  can  .013  be  subtracted  from  125.78, 
and  by  what  must  the  remainder  be  divided  to  give  350,000 
as  a  quotient  ? 

83.  What  length  of  a  road  32  feet  wide  will  have  an  area 
of  half  an  acre  ? 

84.  Find  the  weight  in  tons  per  acre  of  a  rainfall  of  an 
inch,  one  cubic  foot  of  water  weighing  62.5  pounds. 

85.  If  the  gas  for  5  burners,  5  hours  each  evening  for  10 
days,  costs  $1,  what  will  be  the  cost  of  gas  for  75  burners 
which  are  lighted  4  hours  every  evening  for  15  evenings  ? 

86.  The  average  age  of  200  boys  is  14.75  years  ;  what  will 
be  the  average  if  10  boys  are  added  whose  average  age  is  11.6 
years  ? 

87.  The  combined  weight  of  2  bars  of  silver  is  271b.  3  pwt. 
5  gr.  The  larger  one  w^eighs  12  lb.  19  cwt.  21  gr.  more 
than  the  smaller  one.     Required  the  weight  of  each. 

88.  A  and  B  each  have  $10  so  invested  that  A  receives  4 
per  cent  and  B  5  per  cent  interest.  What  per  cent  of  A^s  in- 
terest is  B's  ? 

89.  At  the  rate  of  a  ton  of  coal  every  21  days,  what  will  be 
the  cost  of  the  coal  used  by  a  family  from  Oct.  17,  1904,  to 
April  25,  1905,  excluding  both  of  the  days  named,  at  $4.50  a 
ton  ? 

90.  A  publisher  sells  to  the  wholesale  trade  40  copies  of  a 


MISCELLANEOUS  PROBLEMS.  381 

book  lit  the  retail  price  of  24  copies.     What  does  he  receive 
wholesale  for  a  book  which  retails  at  $1.50  ? 

91.  A^s  weight  is  f  of  B's,  and  C's  is  as  much  as  A^s  and 
B^s  together  ;  the  sum  of  their  weights  is  490  lb.  How 
mucli  does  each  weigh  ? 

92.  The  duty  on  a  shipment  of  blankets  being  33  cents  a 
pound  and  40^  ad  valorem,  what  is  the  invoice  price,  if  they 
cost  the  importer  $3874,  including  the  duty,  the  total  weight 
of  the  blankets  being  5800  lb.  ? 

93.  If  a  man  buys  100  shares  of  railroad  stock  at  107^,  and 
sells  it  a  month  later  when  quoted  at  llof,  what  is  the  gain, 
if  in  the  meanwhile  the  stock  "paid  a  2^  dividend  ? 


94.  The  first  of  four  cog  wheels  which  work  together  has 
21  cogs,  the  second  has  18,  the  third  15  ;  and  the  number  of 
cogs  that  the  fourth  has  is  to  the  number  the  third  has  as  2 
to  3.  After  how  many  revolutions  of  the  smallest  wheel  will 
they  all  be  in  their  position  at  starting  ? 

95.  What  is  the  area  of  a  circle  20  ft.  in  diameter  ? 

96.  Divide  $34.50  among  6  men  and  11  boys,  giving  each 
boy  .5  of  a  man's  share. 

97.  The  great  pyramid,  whose  base  is  square,  measures  763 
ft.  on  each  side.     How  many  acres  does  it  cover  ? 

98.  To  cover  the  floor  of  a  gentleman's  carriage-house,  it 
took  170  planks  16  ft.  long,  9  inches  wide,  and  2  inches  thick. 
How  much  did  the  planks  cost  at  $12  a  thousand  feet  ? 

99.  How  many  yards  of  carpet,  28  inches  wide,  laid  length- 
wise, will  be  required  to  cover  a  floor  9  ft.  4  in.  wide  and 
18^  ft.  in  length  ? 

100.  A  house  which  cost  $4800  rents  for  $300.  If  the 
taxes  are  2  per  cent,  insurance  f  per  cent,  and  repairs  $5.50 
per  annum,  what  per  cent  is  the  net  income  ? 

101.  There  are  in  the  library  of  a  certain  school  683  books, 
which  number  will  give  23  books  to  each  pupil,  and  16  books 
over.     What  is  the  number  of  pupils  ? 


^R2  SCHOOL  ARITHMETIC. 

102.  If  one  pound  of  zinc  covers  a  square  yard,  and  it  is 
worth  45^  a  pound,  what  will  it  cost  to  line  a  tank  10  ft. 
square  and  5  ft.  deep  ? 

103.  \yhen  it  is  13  o'clock  M.  at  St.  Louis,  90°  15'  15"  W., 
what  is  the  time  at  Kichmond,  77°  20'  4"  W.  ? 

104.  How  many  shingles  will  be  required  to  cover  the  roof 
of  a  building  54  feet  long,  the  two  sides  each  16|-  feet  wide, 
if  one  shingle  covers  a  space  6  in.  square  ? 

105.  What  is  the  amount  of  the  following  bill  ? 

32|-  yd.  of  muslin  at  6f  ct. 
I5J  lb.  of  lard  at  12  lb.  for  $1.00. 
23  lb.  3  oz.  of  butter,  at  25^  a  lb. 
13^  lb.  of  sugar  at  16  lb.  for  $1. 
2  doz.  bananas  at  4  for  a  dime. 

106.  Find  the  cost  of  25  joists  10  in.  wide,  18  ft.  long,  and 
4  in.  thick  at  $35  per  M. 

107.  Edward  and  Peter  hire  a  pasture  for  $14.  Edward 
puts  in  8  horses  ;  Peter  puts  in  50  sheep.  If  21  sheep  eat  as 
much  as  2  horses,  what  must  Edward  pay  ? 

108.  If  a  miller  takes  j\  for  toll,  and  a  bushel  of  wheat 
produces  40  lb.  of  flour,  how  many  bushels  of  wheat  must  be 
taken  to  the  mill  to  obtain  a  barrel  of  flour  ? 

109.  The  valuation  of  the  real  property  of  a  village  is 
$125000.  How  many  mills  on  the  dollar  must  be  levied  to 
net  $475  after  paying  the  collector  5  per  cent  ? 

110.  How  much  money  must  I  put  in  a  bank  which  allows 
4^  interest  on  deposits  in  order  to  receive  $100  at  the  end  of 
9  months  ? 

111.  Ten  horses  and  12  cows  cost  $1160  ;  4  horses  and  7 
cows  cost  $530.     What  is  the  value  of  a  horse  ? 

112.  My  shoemaker  sends  me  a  bill  of  $18  for  a  pair  of 
boots  and  two  pairs  of  shoes.  Some  months  afterwards  he 
sends  me  a  bill  of  $32  for  two  pairs  of  boots  and  three  pairs 
of  shoes.     What  do  the  boots  cost  a  pair  ? 


MISCELLANEOUS   PROBLEMS.  383 

113.  In  a  lot  of  eggs  7  of  the  largest,  or  10  of  the  smallest, 
weigh  a  pound.  When  the  largest  are  worth  15^  a  dozen, 
what  are  the  smallest  worth  ? 

114.  When  I  was  married  I  was  3  times  as  old  as  my  wi>e, 
but  15  years  after  our  marriage  I  was  only  twice  her  age. 
Find  my  age  at  marriage. 

115.  The  sum  of  f  of  A's  money  and  ^  of  BV being  on  in- 
terest for  8  years  at  6  per  cent,  gives  I960  interest.  How 
much  money  has  each  if  J  of  B's  is  3  times  f  of  A's  ? 

116.  Take  the  proportion  8  :  3  =  5  :  1|^.  If  the  second 
and  third  numbers  each  be  increased  by  7,  what  multiplier 
will  be  needed  by  the  first  to  make  a  proportion  ? 

117.  Find  the  diagonal  of  a  room  40  ft.  long,  30  ft.  wide, 
and  12  ft.  high. 

118.  The  end  of  the  minute  hand  of  a  town  clock  passes  over 
30  inches  in  12  minutes.     What  is  the  length  of  the  hand  ? 

119.  What  length  of  rope  will  enable  a  horse  to  graze  on  2 
acres  of  grass  ? 

120.  Out  of  a  piece  of  paper  5  ft.  10  in.  square  is  cut  the 
greatest  possible  circle.  How  many  square  inches  of  paper 
are  cut  away  ? 

121.  How  many  pounds  of  flour  should  I  give  for  $42.30, 
at  the  rate  of  $4.90  a  barrel  ? 

122.  How  many  acres  are  in  a  square  the  diagonal  of 
which  is  20  rods  more  than  either  side  ? 

123.  A  bin  6  ft.  long,  2  ft.  wide,  and  1^  ft.  high  is  filled 
with  oats  worth  40^  a  bushel.  What  is  the  value  of  the 
oats  ? 

124.  Find  the  cost  of  paving  and  curbing  one  mile  of 
street,  the  paving  being  30  feet  wide  and  costing  $2.75  a 
square  yard,  and  each  line  of  curbing  costing  30^  a  linear 
foot. 

125.  Two  men  enter  into  a  partnership,  one  putting  in 
$5000,  the  other  $2000.  The  partner  that  puts  in  the  less 
sum  is  to  receive  $300  extra  from  the  proceeds  for  his  ex- 


384  SCHOOL  ARITHMETIC. 

perience  in  the  business.  They  gain  14725.   What  is  the  share 
of  each  ? 

126.  Kequired  the  area  in  acres  of  a  piece  of  land  .5  of  a 
mile  long  and  .3  of  a  mile  broad. 

127.  A  wine  merchant  mixes  12  gallons  of  wine  worth  $1  a 
gallon  with  5  gallons  of  brandy  worth  $1.50  a  gallon,  and  3 
gallons  of  water  of  no  value.  What  is  the  value  of  one  gallon 
of  the  mixture  ? 

128.  How  much  clover  seed  at  15.50  a  bushel  could  a  man 
buy  for  $1017,  after  deducting  his  commission  of  2^^  on 
amount  paid  for  seed,  and  drayage  at  the  rate  of  one  and  a 
fourth  cents  a  bushel  ? 

129.  A  man  has  14400.  How  much  must  he  borrow  at  4^ 
and  put  with  it  so  that  the  two  sums  invested  in  a  business 
that  pays  12  per  cent  per  annum  may  net  him  a  gain  of  $600 
a  year  ? 

130.  John  can  write  a  page  in  a  minutes,  and  Sam  can  do 
the  same  in  r  minutes.  How  much  can  they  both  write  in  5 
minutes  ? 

131.  In  the  A  class  there  are  twice  as  many  girls  as  boys. 
Each  girl  makes  a  bow  to  every  other  girl,  to  every  boy,  and 
to  the  teacher.  Each  boy  makes  a  bow  to  every  other  boy, 
to  every  girl,  and  to  the  teacher.  In  all  there  are  900  bows 
made.     How  many  boys  in  the  class  ? 

132.  Sold  wheat  on  commission  at  6^,  and  invested  the  net 
proceeds  in  flour  at  4^  commission,  my  whole  commission 
being  $625.     What  was  the  value  of  the  wheat  and  flour  ? 

133.  A  man  invested  a  certain  sum  of  money  in  5^  stock  at 
80,  and  twice  as  much  in  4^  stock.  If  his  income  from  the 
former  is  $300,  and  from  the  latter  1^  times  as  much,  what 
was  the  price  of  a  share  in  the  latter  investment  ? 

134.  If  water  expands  -^-^  in  volume  in  being  heated  from 
the  freezing  point  to  the  boiling  point,  find  the  weight  of  a 
cubic  foot  of  boiling  water,  the  weight  at  freezing  point 
being  62.5  poundg, 


MISCELLANEOUS   PROBLEMS.  385 

136.  If  the  pressure  of  the  air  on  the  surface  of  water  is 
15  lb.  per  square  inch,  and  if  1  cu.  ft.  of  water  weighs  1000 
oz.,  find  the  pressure  per  square  foot  of  a  column  of  water 
100  ft.  deep. 

136.  If  each  person  in  breathing  spoils  the  air  of  a  closed 
room  at  the  rate  of  8  cu.  ft.  a  minute,  how  long  can  the 
windows  and  doors  of  a  schoolroom  be  safely  kept  closed 
when  occupied  by  50  pupils,  if  the  room  is  25  ft.  by  20  ft., 
and  10  ft.  high  ? 

137.  By  raising  the  temperature  of  a  cube  of  iron,  the 
length  of  the  edge  was  increased  6^.  Find  the  ratio  of  in- 
crease in  the  volume  of  the  cube. 

138.  Each  side  of  a  hexagonal  (six-sided)  field  is  20  rd., 
and  the  distance  from  the  center  of  the  field  to  the  middle 
point  of  each  side  is  also  20  rd.  What  is  the  area  of  the 
field? 

139.  During  a  heavy  rainstorm,  a  circular  pond  is  formed 
in  a  circular  field.  If  the  diameter  of  the  field  is  250  rd.  and 
that  of  the  pond  125  rd.,  what  is  the  ratio  of  the  land  area  to 
the  water  area  of  the  field  ? 

140.  If  the  average  velocity  of  a  bullet  is  1342  feet  a  sec- 
ond and  that  of  sound  1122  feet  a  second,  how  much  time 
elapses,  on  a  range  of  3000  feet,  between  the  time  the  bullet 
strikes  and  the  time  the  sound  reaches  the  target  ? 

SUPPLEMENTARY    EXERCISES    (FOR    ADVANCED     CLASSES). 

544.  1.  Is  a  billion  a  million  million  ?    Explain. 

2.  Multiply  789627  by  834,  beginning  at  the  left  to  multiply. 

3.  Reduce  f  to  a  fraction  whose  denominator  is  11. 

4.  Change  |,  f,  -j^,  and  .7  to  fractions  having  a  common 
numerator. 

fofi 

5.  Find  the  value  of  -; — rf  -^  .125. 

25 


386  SCHOOL  ARITHMETIC. 

6.  If  I  receive  a  discount  of  20^,  10,^,  and  bfo,  and  sell  at 
a  discount  of  10^,  6fo,  2^^,  what  is  my  per  cent  of  gain? 

7.  At  noon  the  three  hands — hour,  minute,  and  second — 
of  a  clock  are  together.  At  what  time  will  they  first  be  to- 
gether again  ? 

8.  A  merchant  bought  cloth  at  $3.25  a  yard,  and  after 
keeping  it  6  months  sold  it  at  $3.75  a  yard.  What  was  his 
gain  per  cent,  counting  6^  per  annum  for  the  use  of  money  ? 

9.  Bought  10  bushels  of  corn  and  20  bushels  of  turnips 
for  $11 ;  at  another  time  20  bushels  of  corn  and  10  bushels 
of  turnips  for  113.     What  did  the  corn  cost  a  bushel  ? 

10.  A  man  wishing  to  sell  a  horse  and  a  cow  asked  3  times 
as  much  for  the  horse  as  for  the  cow  ;  but  finding  no  pur- 
chaser, he  reduced  the  price  of  the  horse  20^,  and  the  price 
of  the  cow  10^,  and  sold  both  for  1165.  How  much  did  he 
get  for  the  cow  ? 

11.  If  J  of  an  article  is  sold  for  the  cost  of  -J  of  it,  what  is 
the  rate  of  loss  ? 

12.  A  man  sold  a  pig  and  a  sheep  for  $18,  gaining  25^  on 
the  cost  of  the  sheep,  and  20^  on  that  of  the  pig.  If  f  of 
the  cost  of  the  pig  equaled  f  of  the  cost  of  the  sheep,  what 
was  the  cost  of  each  ? 

13.  Mr.  G  spent  $260  for  apples  at  $1.30  a  bushel.  Re- 
taining a  part  for  his  own  use,  he  sold  the  rest  at  a  profit  of 
40^,  clearing  $13  on  the  entire  cost.  How  many  bushels 
did  he  keep  ? 

14.  If  32  men  have  food  for  5  days,  how  many  men  must 
leave,  so  that  the  food  may  last  the  remaining  men  70  days  ? 

15.  What  is  the  difference  in  area  between  a  square  whose 
diagonal  is  one  foot  and  a  circle  whose  diameter  is  one  foot  ? 

16.  A  dealer  bought  3000  bushels  of  oats  at  $.30  a  bushel, 
and  sold  them  out  by  the  bushel  at  10^  above  cost.  When 
all  had  been  sold,  it  was  "^'ound  that  the  quantity  of  oats  had 
shrunk  2^.  What  per  cent  dii  the  dealer  make  on  the 
investment  ? 


MISCELLANEOUS   PROBLEMS.  387 

17.  After  measuring  f  of  a  mile  with  a  chain  100  feet  long, 
the  surveyor  discovers  a  kink  in  the  chain  which  shortens 
its  length  ^  inch.  How  much  less  than  f  of  a  mile  was  the 
distance  measured  ? 

18.  A  man  sold  two  horses  for  $210 ;  on  one  he  gained 
25^,  on  the  other  he  lost  25^.  How  much  did  he  gain, 
supposing  the  second  horse  cost  f  as  much  as  the  first  ? 

19.  A  merchant  sold  goods  at  20^  gain,  but  had  their  cost 
been  $49  more,  he  would  have  lost  15^  by  selling  at  the 
same  price.     How  much  did  the  goods  cost  him  ? 

20.  Had  an  article  cost  20^  more,  the  gain  would  have 
been  25^  less.     What  was  the  gain  per  cent  ? 

21.  A,  B,  and  C,  having  4  loaves  for  which  A  paid  5^,  B 
8^,  and  C  11^,  eat  3  loaves,  and  sell  the  fourth  to  D  for  24^. 
How  many  cents  should  each  receive  ? 

22.  The  distance  from  the  center  of  a  circle  to  the  middle 
of  a  chord  10  inches  long  is  one  foot.  What  is  the  area  of 
the  circle  ? 

23.  A  man  bequeathed  $9000  to  his  three  sons,  aged  13  yr., 
15  yr.,  and  17  yr.,  in  such  a  manner  that  the  share  of  each, 
placed  at  compound  interest  at  6  per  cent  until  he  arrived  at 
the  age  of  21  years,  should  amount  to  the  same  sum.  Find 
the  share  of  each. 

24.  What  is  the  distance  from  the  lower  corner  to  the  op- 
posite upper  corner  of  a  room  15  ft.  by  12  ft.,  and  10  ft.  high? 

25.  If  f  of  the  time  past  noon  equals  f  of  the  time  to  mid- 
night, what  time  is  it  ? 

26.  Between  2  and  3  o'clock  I  mistook  the  minute  hand 
for  the  hour  hand,  and  consequently  thought  the  time  55 
minutes  earlier  than  it  was.     What  was  the  correct  time  ? 

27.  If  sound  travels  in  still  air  1090  ft.  a  second  when  the 
temperature  is  32°  Fahrenheit,  and  if  the  velocity  increases 
1.1  ft.  for  every  degree  of  increase  in  temperature,  how  far 
off  is  an  explosion  when  the  report  follows  in  8  seconds,  the 
temperature  being  70°  ? 


388  SCHOOL   ARITHMETIC. 

28.  If  it  takes  75  Kg.  of  saltpeter,  12.5  Kg.  of  charcoal, 
and  12.5  Kg.  of  sulphur  to  make  100  Kg.  of  powder,  how 
many  kilograms  of  each  will  be  required  to  make  10,000,000 
cartridges,  each  containing  5  g.  of  powder  ? 

29.  In  digging  a  cellar,  15625  cubic  feet  of  earth  were 
removed.  The  length  is  twice  the  width,  and  the  width  is 
twice  the  depth.     What  are  the  dimensions  ? 

30.  What  is  tlie  width  of  a  doorway  8  feet  high  that  is  just 
wide  enough  to  allow  a  circular  mirror  31.416  feet  in  circum- 
ference to  pass  through  ? 

31.  B  has  a  circular  garden  containing  75  sq.  rd.  What 
is  the  area  of  the  largest  square  garden  he  can  make  in  it  ? 

32.  What  is  the  area  of  a  square  whose  diagonal  is  100 
feet? 

33.  A  boy  weighing  96  lb.  is  seated  on  one  end  of  a  see- 
saw 16  ft.  long,  and  a  boy  weighing  120  lb.  is  seated  on  the 
other  end.  Find  the  distance  of  each  boy  from  the  point  of 
support,  the  lengths  of  the  two  arms  of  the  plank  being  in- 
versely proportional  to  the  weights  at  their  ends. 

34.  How  many  apple  trees  can  be  planted  in  an  orchard 
15  rods  square,  allowing  no  two  to  be  nearer  each  other  than 
1:1^  rods  ? 

35.  A  cube  of  water  1.8  dm.  on  an  edge  weighs  how  many 
kilograms  ? 

36.  B  lends  A  $150  for  6  months,  and  a  year  later  A  lends 
B  $100.  How  long  may  B  keep  the  $100  to  balance  the  use 
of  his  loan  to  A  ? 

37.  Starting  from  Dayton,  I  go  200  miles  due  east,  then 
150  miles  due  south,  then  100  miles  due  west,  then  350  miles 
due  north.  How  far  and  in  what  direction  from  Dayton 
am  I  ? 

38.  How  many  rods  is  it  from  the  center  to  one  corner  of 
a  square  field  containing  20  acres  ? 

39.  What  is  the  difference  between  half  a  cubic  foot  and  a 
cubic  half  foot  ? 


MISCELLANEOUS  PROBLEMS.  389 

40.  If  my  horse  had  cost  me  20^  less,  my  rate  of  gain  in 
selling  him  would  have  been  30^  greater.  What  was  my  gain 
per  cent  ? 

100^  of  the  cost  =  the  cost. 
S0%  of  80^  =  24^  of  cost,  yielded  by  20^  of  cost. 
100^  of  cost  yields  5  x  24^  =  120^  of  cost. 
.-.  the  gain  =  20^. 

41.  A  squirrel  goes  spirally  up  a  cylindrical  post,  making 
a  circuit  in  each  5  feet.  How  many  feet  does  it  travel  if  the 
post  is  20  feet  high  and  6  feet  in  circumference  ? 

42.  A  man  agrees  to  pay  $6000  for  a  lot  in  three  equal 
payments,  including  6^  interest  on  unpaid  money.  What  is 
the  yearly  payment  ? 

Amount  of  $6000  at  6^  compound  interest  for  3  yr.  =  $7146.090. 
$1.00  +  $1.06  +  $1.1236  =  $3.1836,  or  3.1836  x  $1. 
$7146.096  -5-  3.1836  =  $2244.66,  the  payment. 

43.  Two-fifths  of  a  mixture  of  wine  and  water  is  wine  ; 
but  if  10  gallons  of  water  be  added  to  it,  then  only  ^'^^  of  the 
mixture  will  be  wine.  How  many  gallons  of  each  liquid  in 
the  mixture  ? 

44.  In  the  center  of  a  circular  island  100  feet  in  diameter 
stands  a  tree  140  feet  high.  A  line  500  feet  long  will  reach 
from  the  top  of  the  tree  to  the  farther  shore.  What  is  the 
width  of  the  river,  the  island  being  in  the  middle  ? 

45.  What  is  the  length  of  the  shortest  possible  route  by 
which  a  fly  can  crawl  from  a  lower  corner  to  the  opposite 
upper  corner  of  a  room  16  feet  long,  12  feet  wide,  and  8  feet 
higli  ? 

46.  In  a  certain  game  A  can  make  20  points  while  B 
makes  30 ;  B  can  make  20  points  while  C  makes  18.  How 
many  points  can  C  make  while  A  makes  100  ? 

47.  Two  trains  start  at  the  same  time,  one  from  Jackson- 
ville to  Savannah,  the  other  from  Savannah  to  Jacksonville. 
If  they  arrive  at  destinations  1  hour  and  4  hours  after  pass- 
ing, what  are  their  relative  rates  of  running  ? 


APPENDIX 


SUPPLEMENTARY     WORK. 


MENSURATION. 

545.  The  process  of  measuring  lines,  surfaces,  and  solids 
is  called  Mensuration. 

The  principles  of  mensuration  that  apply  to  rectangles,  parallelograms, 
triangles,  trapezoids,  circles,  and  rectangular  solids  have  already  been 
given. 

PRISMS. 

546.  A  Polygon  is  a  plane  figure  bounded  by  straight 
lines. 


Thus,  the  ends  of  the  solids  A,  B,  and  C  are  polygons, 

1.  Are   the  two   ends   of    each   solid   equal  ?     Are   they 
parallel  ?     What  is  the  form  of  the  sides  f 

2.  What  is  the  form  of  the  ends  of  the  solid  A  ?     Of  the 
solid  B  ? 

547.  A  solid  whose  ends  are  equal  and  parallel  polygons 
and  whose  sides  are  parallelograms  is  called  a  Prism. 

Thus,  A,  B,  and  C  are  prisms.     The  solid  B  is  also  a  rectangular  solid. 


MENSURATION.  391 

1.  The  polygons  are  called  bases,  and  the  prisms  are  named 
from  the  form  of  the  bases. 

Thus,  A  is  called  a  triangular  prism;  B,  a  quadrangular  prism. 

2.  What  is  the  prism  whose  bases  are  square  called  ? 

3.  When  the  bases  are  parallelograms,  the  prism  is  called 
a  Parallelopiped;  as  B. 

4.  Is  any  triangular  prism  equal  to  half  of  a  parallelopiped 
of  the  same  altitude  and  of  double  the  base  ?  Is  it  therefore 
equal  to  a  parallelopiped  of  the  same  altitude  and  equal 
base  ? 

(Any  prism  can  be  cut  into  triangular  prisms,  as  in  C.) 

548.  To  find  the  volume  of  a  prism. 

How  to  find  the  volume  of  a  rectangular  solid  or  prism  was 
shown  in  Art.  327.  The  volume  of  any  prism  is  found  in  the 
same  way,  viz. : 

Multiply  the  number  of  cubic  units  in  1  unit  of  length  by 
the  number  of  units  of  length.  Or,  multiply  the  area  of  the 
base  by  the  altitude. 

It  should  be  kept  constantly  in  mind  that  the  number  of  cubic  units 
in  1  unit  of  length  is  the  same  as  the  number  of  square  units  in  the  base. 

1.  What  is  the  volume  of  a  prism  whose  length  is  4  feet, 
and  whose  base  is  a  rectangle  4  inches  by  9  inches  ? 

2.  The  base  of  a  prism  9  feet  long  is  a  triangle,  each  of 
whose  sides  is  3  feet,  and  whose  altitude  is  2.598  feet.  How 
many  cubic  feet  does  it  contain  ? 

3.  A  cord  of  Virginia  pine  weighs  2700  lb.  What  is  the 
weight  of  a  single  piece  of  timber  18  inches  square  and  16 
feet  long  ? 

4.  If  a  cubic  foot  of  rolled  steel  weighs  489.6  lb.,  what  is 
the  weight  of  a  piece  4  inches  square  and  30  feet  long  ? 

6.  Find  the  lateral  surface  of  a  prism  whose  length  is  12 
feet,  and  whose  base  is  a  triangle,  each  of  whose  sides  is  2 
feet. 

The  prism  has  3  equal  sides,  each  12  feet  long  and  2  feet  wide.     The 


392 


SCHOOL   ARITHMETIC. 


area  of  one  side  is  24  sq.  ft. ;  hence  the  surface  of  the  3  sides  is  3  times 
24  sq.  ft.,  or  72  sq.  ft.,  the  lateral  surface. 

The  3  sides  together  are  equal  to  one  rectangle  12  ft.  long  and  6  ft. 
wide.     Make  a  rule  for  finding  the  lateral  surface  of  prisms. 

Note. — The  lateral  surface  is  the  entire  surface  except  that  of  the 
ends. 

6.  Find  the  lateral  surface  of  a  prism  3  feet  square  and  8 
feet  long. 

7.  The  sides  of  a  triangular  prism  are  each  2^  feet,  and 
its  height  is  6  feet.     What  is  the  lateral  surface  ? 

8.  What  is  the  lateral  surface  of  a  pentagonal  (five-sided) 
prism  whose  sides  are  each  18  inches,  and  whose  altitude  is 
14  feet  ? 

THE    CYLINDER. 


What  is  the  form  of  the  solid  C  ?  Of  its  ends  ? 
Are  the  ends  equal  ?     Are  they  parallel  ? 

549.  A  solid  whose  ends  (bases)  are  two  equal 
and  parallel  circles,  and  whose  lateral  surface  is  a 
uniformly  curved  surface,  is  called  a  Cylinder. 
It  is  described  by  revolving  a  rectangle  about 
one  of  its  sides  as  an  axis. 

1.  The  circles  are  called  bases. 

2.  Name  four  objects  that  are  cylinders. 

How  does  the  number  of  square  units  in  the  base  of  the  cylinder  C 
compare  with  the  number  of  cubic  units  in  1  unit  of  length  ? 

Then  may  the  volume  of  a  cylinder  be  found  in  the  same  manner  as 
the  volume  of  a  prism  ? 

1.  Find  the  volume  of  a  cylinder  whose  diameter  is  2  feet, 
and  whose  length  is  10  feet. 

2.  In  form  the  Winchester  bushel  is  a  cylinder,  18|  inches 
in  diameter  and  8  inches  deep.  How  many  cubic  inches 
does  it  contain  ? 

3.  A  well  is  26  feet  deep  and  4  feet  in  diameter.  How 
many  gallons  of  water  are  in  it  when  it  is  half  full  ? 


MENSURATION. 


393 


4.  A   cylindrical   pail   8   inches  in   diameter  will  hold   2 
gallons.     What  is  its  depth  ? 

5.  Take  an  oblong  paper  4  inches 
by  6  inches  and  roll  it  to  form  a 
cylinder.  What  is  the  length  of  the 
cylinder  ?     The  circumference  ? 

6.  If  a  hollow  cylinder  is  cut  and  spread  into  a  flat  surface, 
wliat  form  has  it  ? 

7.  What  dimension  of  the  cylinder  is  equal  to  the  length 
of  the  rectangle  ?     What  to  the  width  ? 

8.  Then  how  may  the  lateral  surface  of  the  cylinder  be 
found  ? 

9.  What  is  the  lateral  surface  of  a  cylinder  whose  diameter 
is  2  feet  and  whose  length  is  6  feet  ? 

10.  How  many  square  feet  of  material  in  a  piece  of  stove- 
pipe 6  inches  in  diameter  and  2  feet  8  inches  in  length  ? 

11.  How  many  square  feet  of  tin  will  be  required  to  make 
100  feet  of  spouting  2-^  inches  in  diameter  ? 


THE    CONE    AND    THE    PYRAMID. 


What  is  the  form  of  the  base  of  the  solid  ABC  ?     Notice 
how  the  solid  tapers  to  a  point. 

550.  A  solid  that  tapers  uniformly  from 
a  circular  base  to  a  point  is  called  a  Cone. 
It  is  described  by  revolving  a  right-angled 
triangle  about  one  of  its  sides  as  an  axis. 

1.  The  point  is  called  the  vertex  ;  as  C. 

2.  The  distance  from  the  vertex  to  the 
center  of  the  base  is  the  altitude  ;  as  CO. 

3.  The  shortest  distance  from  the  vertex 

to  the  circumference  of  the  base  is  called  the  slant  height; 
as  CA. 

4.  Name  several  objects  that  have   the  form  of   a  cone. 
Make  a  paper  cylinder  and  a  paper  cone  of  equal  base  and 


394 


SCHOOL  ARITHMETIC. 


altitude.     Fill  the  cone  with  salt,   and  empty  it  into   the 
cylinder.     How  many  conefuls  will  fill  the  cylinder  ? 

It  is  shown  in  geometry  that 

551.  A  cone  has  one  third  the  volume  of  a  cylinder  of  the 
same  base  and  altitude. 

Then  how  may  the  volume  of  a  cone  be  found  ? 

1.  What  is  the  volume  of  a  cone  whose  base  is  12  feet  in 
diameter,  and  whose  altitude  is  18  feet  ? 

2.  Find  the  solid  contents  of  a  cone,  the  diameter  of  whose 
base  is  10  feet,  and  whose  height  is  1 5  feet. 

3.  A  conical  pile  of  grain  is  3  feet  high,  and  the  diameter 
of  its  base  is  6  feet.     How  many  bushels  in  the  pile  ? 

4.  If  a  cubic  yard  of  granite  weighs  4700  lb.,  what  is  the 
weight  of  a  granite  cone  6  feet  high,  the  diameter  of  the 
base  being  4  feet  ? 

The  slant  height  is  the  hypotenuse  of  the  right-angled  triangle  re- 
volved to  describe  the  cone,  and  the  radius  of  the  cone's  base  is  the  base 
of  the  triangle.  (See  figure.)  If  the  slant  height  and  radius  of  base  are 
known,  how  can  the  altitude  be  found  ? 

5.  If  the  slant  height  of  a  cone  is  10  feet,  and  the  diam- 
eter of  the  base  is  12  feet,  what  is  the  altitude  ?  The  vol- 
ume ? 

What  form  has  the  base  of    the  solid  E 

ABCDE  ? 

What  form  has  each  side  ?  Where  do 
the  sides  meet  ? 

552.  A  solid  whose  base  is  a  polygon 
and  whose  sides  are  triangles  meeting  in 
a  point  is  called  a  Pyramid. 

1.  The  point  in  which  the  sides  meet  ^ 
is  the  vertex  ;  as  E. 

2.  The  distance  from  the  vertex  to  the 
centre  of  the  base  is  the  altitude  :  as  EO. 


MENSURATION. 


396 


3.  The  distance  from  the  vertex  to  the  middle  of  a  side  of 
the  base  is  the  slant  height. 

4.  Make  a  paper  prism  and  a  paper  pyramid  of  the  same 
base  and  altitude  from  pasteboard.  If  the  latter  is  filled 
three  times  with  salt,  and  the  contents  poured  into  the  prism, 
will  the  latter  be  exactly  full  ? 

It  is  shown  in  geometry  that 

553.  A  pyramid  has  one  third  the  volume  of  a  prism  of 
the  same  base  and  altitude. 

Then  how  may  the  volume  of  a  pyramid  be  found  ? 

1.  What  is  the  volume  of  a  pyramid  whose  height  is  15 
feet,  and  whose  base  is  8  feet  square  ? 

2.  Find  the  solid  contents  of  a  pyramid  whose  altitude  is 
20  feet,  and  whose  base  is  a  rectangle  12  feet  by  8  feet. 

3.  Find  the  lateral  surface  of  a  pentagonal  pyramid  whose 
slant  height  is  20  feet,  each  side  of  the  base  being  8  feet. 

(a).  The  lateral  surface  is  com- 
posed of  five  equal  sides,  each  a  tri- 
angle whose  dimensions  are  given. 
Thus,  in  the  triangle  ACD,  the 
base  CD  is  8  feet,  and  the  slant 
height  AH  is  20  feet. 

(b).  Since  the  triangles  have  the 
same  altitude,  they  are  together 
equal  to  one  triangle  whose  altitude 
is  20  feet,  and  whose  base  is  40 
feet,  the  perimeter  of  the  base  of 
the  pyramid.  Hence  the  lateral 
surface  =  perimeter  of  base  x  ^ 
slant  height. 

(c).  If  the  number  of  the  sides  of 
the  pyramid  be  increased  indefi- 
nitely, the  bases  of  the  triangles 
will  become  extremely  small,  and 

the  perimeter  may  be  regarded  as  the  circumference  of  the  base  of  a  cone 
whose  lateral  surface  is  equal  to  that  of  the  pyramid.     Hence  the  lateral 


396  SCHOOL   ARITHMETIC. 

surface    of  a  cone  =  circumference   of 
base  X  i  slant  height. 

(d).  If  the  lateral  surface  of  a  cone 
be  imagined  as  unrolled  from  the  solid 
itself,  it  will  appear  as  a  portion  (sector) 
of  a  circle,  as  shown  in  the  figure.  Its 
area  is  equal  to  that  of  a  triangle  whose 
base  equals  the  arc  of  the  sector  and 
whose  altitude  is  the  radius  of  the 
circle.  The  altitude  VD  is  the  slant 
height  of  the  cone.  What  is  the  base  g' 
ADE? 

554.  The  lateral  surface  of  a  cone  =  circumference  of  base 
X  I  sla7it  height. 

1.  Find  the  lateral  surface  of  a  pyramid  whose  base  is  12 
feet  square,  and  whose  slant  height  is  20  feet. 

2.  The  slant  height  of  a  pentagonal  pyramid  is  9  feet,  and 
each  side  of  the  base  is  2  feet.     What  is  the  lateral  surface  ? 

3.  Find  the  lateral  surface  of  a  cone  whose  diameter  at  the 
base  is  16  feet,  and  wliose  slant  height  is  24  feet. 

4.  The  circumference  of  the  base  of  a  cone  is  40  feet,  and 
the  slant  height  is  20  feet.     What  is  the  lateral  surface  ? 

5.  The  cupola  of  a  building  is  16  feet  in  diameter  at  the 
base,  and  measures  22  feet  from  tlie  vertex  to  the  circumfer- 
ence of  the  base.  What  will  be  the  cost  of  painting  it  at 
$.35  a  square  yard  ? 

6.  How  many  yards  of  canvas,  54  inches  wide,  must  be 
bought  to  make  a  conical  tent  having  a  slant  height  of  11 
feet,  and  a  circumference  at  base  of  27  feet  ? 

7.  The  base  of  the  pyramid  Cheops  in  Egypt  is  763.4  feet 
square,  and  the  slant  height  is  612  feet.  How  many  acres  of 
surface  in  its  sides  ? 

8.  Find  the  weight  of  each  of  the  following,  a  cubic  foot 
of  marble  weighing  170  pounds  : 

(a).  A  marble  cylinder — lengtli  3  feet,  diameter  1  foot, 
(b).  A  marble  prism — length  3  feet,  base  1  square  foot. 


MENSURATION.  397 

(c).  A  marble  cone — height  3  feet,  radius  of  base  1  foot, 
(d).  A  marble  pyramid — height  3  feet,  base  1  foot  square. 

THE    SPHERE. 

555.  A  solid  bounded  by  a  surface  whose  every  point  is 
equidistant  from  a  point  within,  called  the  center,  is  a 
Sphere. 

(a).  Bisect  a  sphere  and  observe  the  two  surfaces  exposed.  These  are 
called  great  circles.  Is  the  diameter  of  these  circles  also  the  diameter 
of  the  sphere  ? 

(b).  The  curved  surface  of  a 
hemisphere  is  larger  than  its  flat 
surface.  Is  it  twice  as  large  ?  Take 
a  wooden  hemisphere  and  investi- 
gate by  winding  the  surface  of  the 
hemisphere  with  a  waxed  cord,  and  then  winding  a  great  circle  of  the 
sphere  with  the  cord. 

(c).  It  is  proved  in  geometry  that  the  curved  surface  of  a  hemisphere 
is  twice  the  flat  surface.  If  the  radius  of  the  great  circle  is  r,  the  area  is 
Trr'.     Then  what  is  the  area  of  tlie  curved  surface  ? 

(d).  How  many  great  circles  in  the  curved  surface  of  two  hemispheres, 
or  one  sphere  ?    Then  the  surface  of  a  sphere  =  how  many  times  irr*  ? 

The  area  of  a  circle  =  nr^. 
The  surface  of  a  sphere  =  4;rr'. 

1.  How  many  square  inches  in  the  surface  of  an  8-inch 
globe  ? 

2.  If  the  diameter  of  the  moon  is  reckoned  at  2000  miles, 
how  many  square  miles  in  its  surface  ? 

3.  If  the  diameter  of  the  earth  is  reckoned  at  8000  miles, 
its  area  is  how  many  times  that  of  the  moon  ? 

4.  The  surface  of  a  sphere  is  64  square  feet.  What  is  its 
diameter  ? 

5.  The  area  of  a  great  circle  of  a  sphere  is  100  square 
inches.  Find  the  cost  of  gilding  the  sphere  at  75^  a  square 
foot. 

If  we  join  three  points  on  a  sphere  with  the  center,  we  mark  out  a 


398  SCHOOL  ARITHMETIC. 

solid  which  is  nearly  a  pyramid.  A  sphere  may  be 
regarded  as  made  up  of  a  very  great  number  of  such 
pyramids  whose  common  vertex  is  the  center  of  the 
sphere,  and  whose  bases  are  small  portions  of  the  sur- 
face. The  altitude  of  each  pyramid  is  the  radius  of 
the  sphere,  and  the  area  of  all  the  bases  is  equal  to 
the  surface  of  the  sphere.  Investigate  this  by  cutting 
a  sphere  Can  apple  will  do)  into  pyramids. 

Volume  of  pyramid  =  area  of  base  x  ^  altitude. 
Volume  of  sphere  =  surface  x  ^  radius. 

"       =  4:7tr'  X  ir  =  i7rr\ 

6.  What  is  the  volume  of  a  sphere  whose  diameter  is  12 
inches  ? 

7.  How  many  cubic  inches  in  a  rubber  ball  if  its  diameter 
is  2  inches  ? 

8.  The  diameter  of  a  cannon  ball  is  6  inches.  If  the 
specific  gravity  of  iron  is  7.48,  what  is  the  balFs  weight  ? 

9.  What  is  the  diameter  of  the  largest  sphere  that  can  be 
cut  out  of  a  cube  whose  edge  is  10  inches  ?  What  is  the 
volume  ?     What  per  cent  of  the  cube  is  cut  away  ? 

10.  Find  the  number  of  cubic  miles  in  the  earth,  consider- 
ing the  distance  from  the  surface  to  the  center  as  4000  miles. 

11.  A  12-inch  shell  has  an  inside  diameter  of  10  inches. 
How  many  cubic  inches  of  iron  were  used  in  casting  it  ? 

(From  the  entire  volume  subtract  the  volume  of  the  hollow  portion.) 

SIMILAR     FIGURES. 

556.  Figures  that  have  the  same  shape  are  called  similar 
figures. 

Thus,  lines,  squares,  triangles  whose  angles  are  respectively  equal, 
circles,  cubes,  or  spheres  are  similar  figures.  May  similar  figures  be 
regarded  as  enlarged  or  reduced  copies  of  one  another  ? 

557.  It  is  proved  in  geometry  that — 

1.  The  corresponding  lines  of  similar  figures  are  propor- 
tional. 


MENSURATION.  399 

2.  The  surfaces  {areas)  of  similar  figures  are  to  each  other 
as  the  squares  of  their  corresponding  dimensions.  Conversely, 
their  corresponding  dimensions  are  to  each  other  as  the  square 
roots  of  their  surfaces. 

3.  The  volumes  of  similar  figures  dre  to  each  other  as  the 
cubes  of  their  corresponding  dimensions.  Conversely,  their 
corresponding  dimensions  are  to  each  other  as  the  cube  roots 
of  their  volumes. 

1.  If  the  side  of  one  square  is  twice  that  of  another,  is  its 
area  four  times  as  great  ?  If  the  edge  of  one  cube  is  twice 
that  of  another  cube,  is  its  volume  eight  times  as  great  ?  Il- 
lustrate by  drawing  figures. 

2.  Prove  that  the  surfaces  of  two  spheres  are  to  each  other 
as  the  squares  of  their  radii. 

Let  S  and  s  represent  the  surfaces  of  two  spheres,  and  li  and  r  the 
radii.     Then, 

S  =  ^vR\  and  s  =  4irr'. 

.-. -^  =  49ri2»-i-4irr»  =  :^, 
s  r' 

or  S  :  s=  R""  :  r\ 

3.  Show  that  the  circumferences  of  two  spheres  are  to 
each  other  as  the  radii,  and  the  volumes  as  the  cubes  of 
the  radii. 

4.  One  rectangular  field  containing  12.15  acres  is  36  rods 
long  and  18  rods  wide ;  another  field  of  the  same  shape  is 
27  rods  wide.     Find  the  length  and  area  of  the  larger  field. 

(a)  18    :  27  =  36  rd.  :  x  rd. 

(b)  IS''  :  27'  =  12.15  A.  :  x  A. 

5.  The  weight  of  a  ball  whose  diameter  is  5  inches  is  27 
lb.,  and  the  weight  of  a  similar  ball  is  64  lb.  What  is  the 
diameter  of  the  larger  ball  ? 

5  in.  :  X  in.  =    -^27  :    ^6l  =8:4. 
.*.  the  required  diameter  =  6f  in. 

6.  How  many  circles  2  inches  in  diameter  are  equal  to  a 
circle  whose  diameter  is  6  inches  ? 


400  SCHOOL  ARITHMETIC. 

7.  An  8-inch  square  is  equal  to  how  many  2-inch  squares  ? 

8.  In  a  park  are  two  circular  flower  beds,  one  three  times 
as  large  as  the  other.  Find  the  circumference  of  the  larger, 
if  the  smaller  is  25  feet  ? 

9.  A  rectangular  lot*  is  20  rods  long  and  4  rods  wide.  A 
similar  lot  contains  2^  acres,  and  is  surrounded  by  a  fence 
which  cost  $1.75  a  rod.     Find  the  cost  of  fencing  it. 

10.  A  cow  is  tied  to  a  stake  by  a  rope  9  yards  long,  and  a 
horse  is  tied  to  another  stake  by  a  rope  6  yards  in  length. 
Upon  how  much  more  area  can  the  cow  graze  than  the 
horse  ? 

11.  If  25  gallons  of  water  flow  through  a  pipe  2  inches  in 
diameter  in  a  minute,  how  many  gallons  an  hour  will  flow 
through  a  pipe  6  inches  in  diameter  ? 

12.  If  a  pipe  1  ft.  6  in.  in  diameter  fills  a  cistern  in  6 
hours,  what  is  the  diameter  of  a  pipe  that  will  fill  it  in  1  hr. 
30  min.? 

13.  How  many  square  feet  of  zinc  will  be  required  to  line 
the  sides  and  bottom  of  a  cubical  box  whose  capacity  is  equal 
to  that  of  a  rectangular  box  4  ft.  6  in.  long,  3  ft.  3  in.  wide, 
and  2  ft.  1^  in.  deep  ? 

14.  A  cubical  box  is  2  ft.  deep.  What  is  the  depth  of  an- 
other cubical  box  that  holds  three  times  as  much  ? 

15.  A  pail  9  inches  deep  will  hold  2  gallons.  What  is  the 
depth  of  a  similar  pail  that  holds  2  quarts  ? 

16.  The  diameter  of  one  cannon  ball  is  2J  times  that  of 
another,  which  weighs  27  lb.  What  is  the  larger  ball  worth 
at  1^  a  pound  ? 

17.  What  is  the  edge  of  a  cube  whose  contents  are  equal 
to  the  contents  of  two  cubes  whose  edges  are  respectively  3 
feet  and  5  feet  ? 

18.  The  Winchester  bushel  is  18J  inches  in  diameter  and 
8  inches  deep.  What  are  the  dimensions  of  a  similar  measure 
that  holds  ^  peck  ? 


MENSURATION.  401 


SUPPLEMENTARY    PROBLEMS. 

668.  1.  The  altitude  of  a  pyramid  is  12  feet,  and  its  base 
is  18  feet  square.  What  will  be  the  cost  of  painting  the 
lateral  surface  at  $.45  a  square  yard  ? 

2.  If  a  3-inch  pipe  fills  a  cistern  in  9J^  hours,  how  large  a 
pipe  will  fill  it  in  12  hours  ? 

3.  How  many  2-inch  balls  can  be  made  from  a  ball  6 
inches  in  diameter  ? 

4.  A  box  has  a  bottom  2  ft.  6  in.  square,  the  top  is  3  ft.  6 
in.  square,  the  height  is  2  ft.  6  in.  What  will  it  cost  to  line 
with  zinc  at  20^  a  square  foot  ? 

5.  What  is  the  side  of  the  largest  cube  that  can  be  cut 
from  a  sphere  17  inches  in  diameter  ? 

6.  If  a  pipe  1.5  inches  in  diameter  fills  a  cistern  in  5  hours, 
in  what  time  will  another  whose  diameter  is  15  inches  fill 
it  ? 

7.  A  conical  candle  is  one  inch  thick  at  the  bottom,  and 
burns  away  the  first  inch  in  15  minutes  ;  it  continues  to  burn 
at  the  same  rate,  and  is  consumed  in  54  hours.  Find  its 
length. 

8.  If  a  4|-inch  pipe  fills  a  cistern  in  5^  hours,  how  long 
will  it  take  a  3-inch  pipe  to  fill  it  ? 

9.  How  many  half-inch  bullets  can  be  made  from  a  lead 
ball  5  inches  in  diameter  ? 

10.  A  cubic  foot  of  brass  is  to  be  drawn  into  a  wire  ^  of 
an  inch  in  diameter.     What  will  be  the  length  of  the  wire  ? 

11.  How  many  inch-pipes  will  be  required  to  empty  a 
reservoir  as  fast  as  a  foot-pipe  fills  it  ? 

12.  The  sides  of  3  regular  octagons  are  3  ft.,  4  ft.,  and  12 
ft.,  respectively.  Find  the  side  of  a  fourth  octagon  whose 
area  is  equal  to  that  of  the  first  three. 

13.  How  many  square  yards  of  cloth  will  be  required  to 
make  a  conical  tent  10  ft.  in  diameter  and  12^  ft.  high  ? 

14.  The  diameter  of  the  earth  is  about  4  times  that  of  the 

36 


402  SCHOOL   ARITHMETIC. 

moon.  How  many  moons  should  weigh  as  much  as  the  earth, 
assuming  them  to  be  composed  of  like  material  ? 

15.  A  conical  wine  glass  2  inches  in  diameter  and  3  inches 
deep  is  ^  full  of  water.     What  is  the  depth  of  the  water  ? 

16.  Three  men  bought  a  grindstone  3  feet  in  diameter. 
How  much  of  the  diameter  must  each  grind  off  to  use  up  his 
share  of  the  stone,  making  no  allowance  for  the  eye,  or 
aperture  ? 

17.  Four  women  bought  a  ball  of  yarn  6  inches  in  diam- 
eter, and  agreed  that  each  should  take  her  share  in  turn  by 
winding  from  the  outer  part  of  the  ball.  How  much  of  the 
diameter  did  each  wind  off  ? 

18.  The  number  of  oscillations  that  pendulums  make  in  a 
given  time  is  inversely  as  the  square  root  of  the  numbers 
representing  their  lengths.  The  length  of  a  1-second  pendu- 
lum being  .994  m.,  what  is  the  length  of  a  pendulum  that 
beats  half-seconds  ? 

19.  A  cylinder  12  feet  in  diameter  is  equivalent  to  a  cone 
18  feet  in  diameter  and  8  feet  high.  What  is  the  height  of 
the  cylinder  ? 

20.  Two  circular  plates  of  the  same  thickness  and  material 
have  diameters,  the  one  7  inches,  the  other  x  inches.  If  the 
weight  of  the  latter  is  40^  of  the  former,  find  the  value  of  x. 

21.  If  a  ball  of  yarn  4  inches  in  diameter  makes  one  pair 
of  gloves,  how  many  similar  pairs  will  a  ball  8  inches  in 
diameter  make  ? 

22.  Find  the  amount  of  tin  necessary  to  make  a  tin  pail 
6  inches  in  diameter  and  8  inches  deep. 

23.  A  hollow  sphere  8  inches  in  diameter  is  filled  with 
water.  How  many  hollow  cones,  each  8  inches  in  altitude, 
and  8  inches  in  diameter  at  the  base,  can  be  filled  with  the 
water  in  the  sphere  ? 

24.  A  cylindrical  tank  is  1.2  m.  in  diameter  and  3  m.  long. 
If  it  is  full  of  petroleum,  which  is  .7  as  heavy  as  water,  what 
is  the  weight  of  the  petroleum  ? 


MENSURATION.  408 

26.  Find  the  dimensions  of  a  cylinder,  having  its  diameter 
equal  to  its  height,  that  will  hold  1  liter. 

26.  The  volume  of  a  cone  is  1  cu  m.  What  are  its  dimen- 
sions if  its  height  is  equal  to  the  radius  of  its  base  ? 

27.  If  the  air  around  the  earth  is  40  miles  deep,  and  the 
diameter  of  the  earth  is  taken  as  7920  miles,  how  many  cubic 
miles  of  air  are  there  ? 

28.  Find  the  radius  of  that  sphere  the  number  of  square 
centimeters  of  whose  surface  is  three  times  the  number  of 
cubic  centimeters  of  its  volume. 

29.  A  hollow  sphere  is  32  cm.  in  diameter,  and  the  shell 
38  mm.  thick.  If  the  weight  of  the  metal  is  7.2  as  heavy  as 
water,  what  is  the  weight  of  the  sphere  ?  How  much  will  it 
hold  ? 

30.  A  trapezoidal  board  12  feet  long  is  16  inches  wide  at 
one  end,  and  8  inches  at  the  other.  How  far  from  either  end 
must  it  be  cut  so  that  each  part  may  contain  one  half  of  it  ? 


f 


Since  the  board,  represented   in  the  figure  by  A  BOD, 
decreases  8  in.  in  12  ft.,  its  non-parallel  sides  will  meet  in  ; 

a  point  in  24  ft.,  if  imagined  to  be  produced  as  indicated  / 

by  the  dotted  lines.  / 

.-.  area  of  ABG  =  i  x  24  x  1^  x  1  sq.  ft.  =  16  sq.  ft. 

Area  of  board  ABCD  =  12  sq.  ft.     Why  ? 

.-.  area  of  EFG  =  16  sq.  ft.  -  6  sq.  ft.  =  10  sq.  ft. 

From  similar  triangles,  16  :  10  =  24''  :  GN'. 

Hence  GN  =   V'SQO  =:  18.97  +  . 

.-.  18.97  ft.  -  13  ft.  =  6.97  ft.,  MN,  the  distance  from 
the  narrow  end. 


31.  A  pole  120  ft.  long  breaks  so  that  the  top  touches  the 
ground  40  ft.  from  the  foot.  What  is  the  height  of  the 
stump  ? 

Let  AT  represent  the  pole,  C  the  place  it  breaks,  and  B  the  point 
where  T  touches  the  ground. 


404 


SCHOOL  ARITHMETIC. 


AT  =  120  ft.,  AB  =  40  ft. 

Construct  the  squares  on  BC  and  CE,  CE 
being  taken  equal  to  AC. 

Since  P  is  equal  to  the  square  on  AC,  it  is 
evident  that  the  square  on  the  hypotenuse  ex- 
ceeds that  on  the  perpendicular  by  the  two 
rectangles^  and  h,  whose  combined  length  is 
120  ft. ,  and  whose  area  is  equal  to  the  area  of 
the  square  on  AB,  or  1600  sq.  ft. 


1600  -4-  120  =  VM 


the  common  width 


i  (130  ft.  -  13^  ft.),    or 


of  the  rectangles  j9  and  h  is  13^  ft.  But  this 
width  is  the  difference  in  the  lengths  of  AC 
and  BC.  Their  sum  =  120  ft.  Hence,  i 
(120  ft.  +  13^  ft.),  or  66f  ft.,  =  BC  ;  and 
53i  ft,  =  AC. 

32.  A  limekiln  measured  at  the  bottom  50  ft.  long  and  20 
ft.  wide  ;  at  the  top  40  ft.  long  and  16  ft.  wide.  The  height 
was  6  ft.     How  many  cubic  feet  of  lime  in  the  kiln  ? 

The  limekiln  may  be  regarded  as  the  frustum  of  a  pyramid.  The 
volume  of  the  frustum  of  a  pyramid  (or  cone)  of  bases  B,  b,  and  altitude 
h,  as  shown  in  geometry,  is  expressed  by  the  formula 


V^'liB  +  h  ^ 


VBh). 


The  area  of  the  lower  base  =  20  x  50  sq.  ft.  =  1000  sq.  ft. 
The  area  of  the  upper  base  =  16  x  40  sq.  ft.  =    640  sq.  ft. 
Hence  the  volume  =  f  (1000  +  640  +    VlOOO  x  640)   cu.  ft.  =  4880 
cu.  ft. 

GREATEST    COMMON    DIVISOR. 


559.  The  Greatest  Coniinoii  Divisor  of  two  or  more 
numbers  is  the  greatest  numher  that  exactly  divides  each  of 
them. 

Thus,  7  is  the  greatest  common  divisor  of  21,  35,  and  42.     Why  ? 
1.  Find  the  G.  C.  D.  of  56,  84,  and  140  by  the  method  of 
factoring. 
56  =:  2"    X   2   X   7.  Since  the  greatest  common  divisor  is  the 

84  =  2^*    X   3   X    7.     greatest  factor  common  to  the  three  numbers, 
140  =  2'   X   5   X  7.     it  is  2"  X  7,  or  28. 


GREATEST  COMMON   DIVISOR.  405 

2.  Find,  by  factoring,  the  G.  C.  D.  of  : 

(a).  32,  48,  128.     (b).  84,  12G,  128.     (c).  187,  253,  341. 

560.  To  find  the  greatest  common  divisor  when  the  num- 
bers cannot  be  easily  factored,  tlie  long  division  process  (also 
called  the  **  Euclidean  method,"  from  Euclid,  who  first  used 
it)  is  usually  employed.     This  method  depends   upon    two 

principles  : 

1.  A  factor  of  a  number  is  a  factor  of  any  of  its  multiples. 

2.  Every  common  factor  of  ttvo  numbers  is  also  a  factor  of 
their  sum  and  of  their  difference. 

Thus,  7,  which  is  a  factor  of  14,  is  also  a  factor  of  28,  35,  etc.  ;  and, 
being  a  common  factor  of  35  and  126,  it  is  also  a  factor  of  161,  their  «Mm, 
and  of  91,  their  difference. 

1.  Find  the  G.  C.  D.  of  63  and  231  by  the  method  of 
continued  division. 

63  )231(  3  Since  21  is  a  factor  of  itself  and  of  42,  it  is 

189  a  factor  of  63,  their  sum. 

~~7k  \cn/  1  Since  21  is  a  factor  of  6),  it  is  a   factor  of 

'^2  ^  ^  63,  or  189,   a  multiple  of  03;  and  being  a 

—  factor  of  42  and  of  189,  it  is  a  factor  of  42  +  189, 

21)42(2       or  231,  their  sww. 

^  .-.  21  is  a  common  factor  of  63  and  231. 

Again,  every  common  factor  of  63  and  231  is  a  factor  of  3  x  63,  or 
189,  a  multiple  of  63  ;  and  also  a  factor  of  231  —  189,  or  42,  their  dif- 
ference. 

Since  every  such  factor  is  now  a  common  factor  of  63  and  42,  it  is  a 
factor  of  63  —  42,  or  21,  their  difference. 

Since  the  greatest  common  factor  of  63  and  231  is  contained  in  21,  it 
cannot  be  greater  than  21. 

.-.  21  is  the  G.  C.  D.  of  63  and  231. 
Find  by  the  last  method  the  G.  C.  D.  of  : 

{a).  9801  and  33759.  {b).  3864,  3404,  and  3657.  {c). 
4656  and  5926. 

To  find  the  G.  C.  D.  of  several  numbers  by  this  method,  find  the 
G.  C.  D.  of  two  of  them  ;  then  of  that  result  and  a  third  number,  and 
so  on. 


406  SCHOOL   ARITHMETIC. 


COMPOUND   PROPORTION. 

661.  The  product  of  two  or  more  ratios  is  called  a  Com- 
pound Ratio. 

Thus,  3  X  5  :  4  X  7,  or  15  :  28,  is  the  compound  ratio  of  3  :  4  and 
5:7.' 

562.  Any  equation  each  member  of  which  is  composed  of 
two  factors  may  be  written  as  a  proportion. 

Take  the  simple  proportion  5  :  8  =  15  :  24.  Solving,  we  have, 
5  X  24  —  8  X  15.  It  will  be  observed  that  one  member  contains  the 
extremes  of  the  proportion,  the  other  the  means.  The  equation  ah  =  cd 
may  be  written, 

a  :  c  =  d  :  b, 
ov,  a  :  d  =  c  :  b. 

The  compound  ratio  of  a  :  c  and  a  :  d  is  a  x  a  :  c  x  d  ;  that  of  d  :  b  and 
c  :  b  is  d  X  c  :  b  X  b.  And  since  the  simple  ratios  of  each  proportion 
are  equal,  their  products  are  equal.     Hence 

axa:  cxd  =  dxc:  bxb. 
That  is,  one  compound  ratio  is  equal  to  the  other. 

563.  An  expression  of  the  equality  of  two  compound 
ratios,  or  of  a  compound  ratio  and  a  simple  one,  is  called  a 
Compound  Proportion. 

Thus,  the  equation  f  x  f  =  ^^  may  be  written  in  the  form  of  the 
compound  proportion 

^  •  ^  =  10  :  28 

3  :7 

By  taking  the  product  the  proportion  is  reduced  to  the  simple  one 
15:42=10:28. 

Note. — In  a  compound  proportion  the  product  of  the  extremes  is 
equal  to  the  product  of  the  means,  as  in  the  case  of  a  simple  proportion; 
hence  the  required  number  is  found  in  the  same  way.  In  problems  of 
this  class  it  is  convenient  to  make  the  number  which  is  of  the  same  kind 
as  the  answer  the  third  one,  and  then  to  consider  each  of  the  remaining 
pairs  of  numbers  separately,  forming  a  first  couplet  from  each,  as  in 
simple  proportion. 


COMPOUND  PROPORTION.  407 


WRITTEN    EXERCISES. 

564.  1.  If  8  horses  eat  48  busliels  of  oats  in  24  days,  in 
how  many  days  will  4  horses  eat  38  bushels  ? 

^^^;|3  =  24days:(    ,. 

.-.  the  required  time  is  8  x  38  x  24  da.   ^  gg  ^^^^ 

4  X  48 

The  problem  is  to  determine  the  ratio  resulting  from  each  compari- 
son, and  how  they  affect  the  number  of  days  which  we  are  required  to 
find.  For  convenience  we  make  the  24  days  the  third  number,  as  the 
answer  is  to  be  in  days. 

It  will  require  more  daya  for  4  horses  to  eat  a  given  quantity  than  for 
8  horses  to  eat  the  same  amount.  Therefore  we  make  4  the  first  number 
and  8  the  second. 

It  will  require  less  time  for  the  same  number  of  horses  to  eat  38  bu. 
than  to  eat  48  bu.  Therefore,  we  make  48  the  first  numl)er  and  38  the 
second.     In  finding  the  fourth  number,  how  may  the  work  be  simplified? 

2.  If  24  men  in  5  days  can  build  a  wall  72  rd.  long,  how 
many  rods  of  wall  can  15  men  build  in  6  days  ? 

3.  If  10  men  can  cut  46  cords  of  wood  in  18  days,  working 
10  hours  a  day,  how  many  cords  can  40  men  cut  in  24  days, 
working  9  hours  a  day  ? 

4.  If  a  railroad  charges  $15  for  carrying  3  tons  of  goods 
180  miles,  what  will  it  cost  at  the  same  rate  to  transport 
15000  lb.  of  goods  140  miles  ? 

5.  If  36  men  earn  $1296  in  18  days,  how  much  will  42  men 
earn  in  87  days  ? 

6.  What  is  the  weight  of  a  block  of  granite  8  feet  long, 
4  feet  wide,  and  10  inches  thick,  if  another  block  10  feet 
long,  5  feet  wide,  and  16  inches  thick  weighs  5200  lb.? 

7.  A  miller  has  a  bin  9  ft.  long,  4  ft.  wide,  and  2  ft.  deep, 
holding  72  bushels  of  wheat.  How  long  must  he  make  an- 
other bin  which  is  to  be  5  ft.  wide  and  4  ft.  deep  in  order 
that  it  may  hold  192  bu.  of  wheat  ? 

8.  If  the  cost  of  digging  a  cellar  36  ft.  long,  25  ft.  wide, 


408  •  SCHOOL  ARITHMETIC. 

and' 6  ft.  deep  is  $90,  what  is  tlie  cost  of  digging  a  cellar  45 
ft.  long,  28  ft.  wide,  and  8  ft.  deep  ? 

9.  If  120  men  in  15  days  can  do  J  of  a  certain  piece  of 
work,  how  many  men  in  30  days  can  do  f^  of  the  same  work  ? 

10.  If  58  men  working  9  hours  a  day  require  6  days  to  dig 
a  trench  100  yd.  long,  2  yd.  wide,  and  3  yd.  deep,  how  many 
men  working  10  hours  a  day  for  9  days  will  be  required  to 
dig  a  trench  50  yd.  long,  6  yd.  wide,  and  5  yd.  deep,  in 
ground  twice  as  hard  to  dig  ? 

INSURANCE. 

065.  1.  A  house  valued  at  $1000  was  insured  against  fire 
for  one  year  at  1^.  What  was  the  premium,  or  cost  of 
insurance  ? 

2.  A  building  worth  $10000  is  insured  for  ^  of  its  value  at 
Ifo.     What  is  the  premium  ? 

3.  If  I  pay  $10  for  having  my  property  insured  at  1^,  for 
what  amount  do  I  get  it  insured  ? 

566.  Insurance  is  a  contract  by  which  one  party  agrees 
to  indemnify  another  for  loss  sustained  in  the  event  of  cer- 
tain misfortunes. 

The  three  most  common  forms  -Are  Fire  insura.nce,  Accide fit  insurance, 
and  Life  insurance. 

567.  The  written  contract  of  insurance  is  called  the 
Policy.  It  contains  a. promise  to  pay  a  specified  sum  in  the 
event  of  certain  contingencies.  This  sum  is  the  face  of  the 
policy. 

Fire  insurance  companies  do  not  usually  insure  more  than  f  of  the 
value  of  a  property. 

568.  The  sum  paid  for  insurance  is  called  the  Premium. 
It  is  usually  computed  at  a  given  sum  for  each  $100  or  $1000 
of  insurance  ;  but  sometimes  at  a  certain  per  cent  of  the  face 
of  the  policy. 


INSURANCE.  409 

WRITTEN     EXERCISES. 

569.  1.  The  owner  of  a  store  insures  for  $16750,  at  75^ 
per  1100.     How  nuicli  is  the  annujil  premium  ? 

2.  Wliat  will  it  cost  to  insure  a  house  for  14800,  at  If^  ? 

3.  AVIiat  will  it  cost  to  insure  a  mill  worth  ILSOOO  for  ^  of 
its  value,  at  11.50  per  $100  ? 

4.  What  is  the  premium  for  insuring  property  against  loss 
by  fire  for  1  yr.  for  $3500  at  $1.20  per  $100,  and  the  con- 
tents for  $7500  at  $1.30  per  $100  ? 

5.  A  house  worth  $8000  was  insured  by  3  companies  for  -j^*^ 
of  its  value.  The  first  took  ^  of  the  risk  at  2^%,  the  second 
^  of  the  risk  at  2^,  and  the  tliird  the  remainder  at  2^^. 
What  was  the  total  premium  ? 

6.  For  how  much  must  property  worth  $21825  be  insured 
at  $3  per  $100,  to  cover  both  property  and  premium  ? 

7.  I  paid  $72  for  insuring  my  house  at  2^.  What  was  the 
face  of  tlie  policy  ? 

8.  I  paid  $60.75  for  insuring  property  worth  $2700.  What 
was  the  rate  of  insurance  ? 

9.  A  grain  shipper  paid  $525  for  the  insurance  of  a  cargo 
of  wheat  at  $1.50  per  $100.  For  how  much  was  the  wheat 
insured  ? 

10.  If  SSfo  of  the  value  of  a  ship  is  insured  at  a  cost  of 
$271.33^,  at  f^  premium,  what  is  the  value  of  the  ship  ? 

11.  A  man  30  years  old  insures  his  life  for  $2500,  at  the 
rate  of  $22.50  for  every  $1000  of  insurance.  What  is  the 
annual  premium  ?     How  much  d   es  he  pay  in  20  years  ? 

12.  A  man  had  his  life  insured  for  $10000  at  $32.40  per 
$1000.  Should  he  die  after  having  paid  18  premiums, 
how  much  more  would  his  heirs  receive  than  he  had  paid  in 
premiums  ? 

13.  A  man  paid  an  insurance  company  for  30  years  an  an- 
nual premium  on  a  life  policy  for  $5000  at  the  rate  of  $2*^.85 
per  $1000.  If  15^  of  this  premium  was  returned  in  divi- 
dends, how  much  did  he  pay  for  his  insurance  ? 


410  SCHOOL  ARITHMETIC. 


EXCHANGE. 


570.  The  system  of  making  payment  of  debts  at  distant 
places  without  the  transmission  of  money  is  called  Ex- 
change. 

571.  If  John  Doe,  of  Baltimore,  owes  Richard  Roe,  of 
Rochester,  $500,  he  can  cancel  the  debt  in  several  ways  with- 
out actually  sending  the  money.  He  can  send  a  chech,  a 
draft,  Q,  postal  money  order,  or  an  express  money  order. 

If  Doe  draws  a  check  for  the  |500  payable  to  the  order  of  Richard  Roe, 
he  sends  it  to  the  latter,  who  indorses  it  and  has  it  cashed.  The  bank 
receiving  it  collects  the  sum  named  from  the  bank  upon  which  it  is  drawn. 

572.  A  Bank  Draft  is  a  written  order  from  one  bank 
directing  another  bank  to  pay  a  specified  sum  of  money  to 
the  order  of  the  person  named  in  the  draft. 

The  following  is  a  common  form  of  draft : 


flDercbante^  matlonal  Banh. 

Baltimore,  ^une  f5,  /poo. 
Pay  to  the  order  of  B^lchaxd  Sfboe $5oo.oo. 

cfive  BSundzed^ ^^DollarS. 

^.  c^.  "Walkct,  Cashier, 

To  the  (S^dtoz  €Bank, 
%e^  york  6iiy. 


John  Doe  may  discharge  his  indebtedness  to  Roe  by  paying  the  money 
to  his  Baltimore  bank,  which  in  that  case  delivers  to  him  a  draft  for  the 
$500  on  a  New  York  bank,  payable  to  his  own  order,  or  to  the  order  of 
Richard  Roe.  This  New  York  draft  Doe  sends  to  Roe,  who  indorses  it 
and  has  it  cashed  as  he  did  the  check. 

For  its  service  the  Baltimore  bank  may  charge  Doe  a  small  sum  (cost 


EXCHANGE.  411 

of  exchange),  probably  25^  or  50^,  or  it  may  be  tV  of  1%  on  the  face  of 
the  draft.  Many  banks  charge  their  patrons  nothing  for  New  York  ex- 
change. 

573.  The  cost  of  the  exchange  is  either  a  merely  nomi- 
nal sum  to  cover  expenses,  or  a  certain  per  cent  of  the  face 
of  the  draft,  usually  r^^. 

574.  If  the  cost  of  the  draft  is  greater  than  the  face, 
exchange  is  said  to  be  at  a  premium  ;  if  less  than  the  face, 
it  is  said  to  be  at  a  discount. 

If  the  banks  of  any  city,  say  New  Orleans,  have  not  sufficient  funds 
on  deposit  in  New  York  to  meet  tlie  drafts  they  are  making  on  that  city, 
they  must  incur  the  expense  of  sending  the  money  to  meet  these  drafts. 
This  raises  the  cost  of  drafts  in  New  Orleans,  and  exclmnge  on  New  York 
is  at  a  premium.  But  if  the  New  Orleans  bankers  have  an  abundance  of 
funds  standing  to  their  credit  in  New  York,  they  sell  drafts  on  that  city 
at  a  discount  in  order  to  get  money  for  use  at  home  without  incurring 
the  expense  and  risk  of  having  it  forwarded  by  express. 

575.  A  Commercial  Draft  is  a  written  order  from  one 
person  to  anotlier,  directing  him  to  pay  a  stated  sum  of  money 
to  the  order  of  the  bank  named  in  the  draft.  Commercial 
drafts  are  extensively  used  by  creditors  to  demand  payment 
and  collect  debts  through  banks. 

The  following  is  a  common  form  : 


Rochester^  N,  V.,  July  f5,  /poo. 
t  Alght  ^ — Pay  to  the 

%ke  cflxAt  National  cBank  of  cBjocheAtex 

cfive    BSundzed j%  DollaVS. 

Shickaxd    Oooe, 


To  Jokn    2)oe, 

p8  ^ood  St.,   malUmotc,   SJBd. 


412  SCHOOL  ARITHMETIC. 

576.  If  a  draft  is  made  payable  on  its  presentation,  it  is 
called  a  sight  draft ;  if  payable  at  a  specified  time  after  sight 
or  after  date,  it  is  called  a  time  draft. 

In  many  states  3  days  of  grace  are  allowed  on  time  drafts, 
and  in  some  states  grace  is  also  allowed  on  sight  drafts. 

1.  If  John  Doe  owes  Richard  Roe  the  $500  and  is  slow  in  paying  it, 
the  latter  may  make  out  a  draft  as  above  and  deposit  it  for  collection. 
The  Rochester  bank  will  then  send  it  to  some  Baltimore  bank,  with  a  re- 
quest to  collect  and  remit.     This  is  called  "  drawing  on  "  a  debtor. 

3.  The  Baltimore  bank  will  present  it  to  John  Doe  and  demand  pay- 
ment. If  a  sight  draft,  he  may  pay  it  on  presentation,  if  a  time  draft 
(or  a  sight  draft  where  grace  is  allowed  on  same),  he  may  write  the  word 
"accepted,"  the  date,  and  his  name  across  the  face.  This  is  called 
"  accepting  the  draft."  He  is  then  responsible  for  its  payment,  but  is 
not  liable  unless  and  until  he  "accepts."  At  the  proper  time  it  will 
again  be  presented  by  the  bank  and  payment  demanded. 

3.  If  he  declines  to  accept  it,  or  to  pay  it  on  presentation,  it  is 
returned  to  the  Rochester  bank  and  Roe  is  notified.  If  Doe  pays  the 
draft,  the  Baltimore  bank  remits  to  the  Rochester  bank,  deducting  a 
small  sum  (cost  of  exchange)  for  making  the  collection. 

4.  An  "accepted"  draft  is  in  effect  a  note  whose  date  is  the  date  of 
acceptance  if  payable  so  many  days  after  date  ;  otherwise  the  date  of  the 
draft  is  the  date  of  the  note. 

577.  The  Postal  Money  Order  is  an  order  drawn  by 
one  postmaster  on  another,  directing  him  to  pay  a  specified 
sum  of  money  to  the  person  named  therein,  or  to  his  order. 
The  fees  charged  (cost  of  exchange)  vary  from  3^  to  30^*,  ac- 
cording to  the  amount. 

578.  The  Express  Money  Order  is  substantially  like 
the  postal  money  order.  The  fees  are  the  same,  except  that 
on  all  orders  not  over  $5  the  fee  is  5^. 

579.  Foreign  Excliang-e  is  subject  to  the  same  general 
laws  as  exchange  between  different  cities  of  this  country — 
domestic  exchange — differing  chiefly  as  to  currency  and 
the  manner  of  making  quotations. 

580.  Foreign  drafts  are  usually  called  Bills  of  Exchange, 


EXCHANGE.  413 

and  are  now  generally  drawn  in  duplicate,  formerly  in  sets  of 
three.     These  are  sent  by  different  mails  to  avoid  loss  or  de- 
lay.    When  one  is  accepted  or  paid,  the  others  are  void. 
The  following  is  the  usual  form  : 


£5oo.  New  York,  o^ay  i,  tpct. 

At  sight  of  this  First  of  Exchange  {Second 
of  same  tenor  and  date  unpaid^  pay  to  the  order 

of  cRyichatd   ffuoe,    ffive   &6undzcd    £Sounda,   ValuC    rC- 

ceived,  and  charge  same  to  account  of 

To  cBaxincf  mtotliexd,  '^  3        i^ 

>on()on,  onc/i 


581.  Exchange  for  sight  drafts  was  quoted  in  New  York, 

on  June  30,  1900,  as  follows  : 

On  London,  at  $4.8G5  for  1  pound  sterling,  meaning  that 
£1  in  gold  was  worth  $4,865  in  gold,  the  exchange  being 
quoted  as  so  many  dollars  to  the  pound. 

On  Paris,  at  5.18  francs  for  $1,  meaning  that  $1  would 
buy  a  draft  for  5.18  francs.  It  is  sometimes  quoted  as  so 
many  cents  to  the  franc. 

On  German  cities,  at  4  reichsmarks  for  $.945,  meaning 
that  $.945  will  buy  a  draft  for  4  marks.  It  is  sometimes 
quoted  as  so  many  cents  to  the  mark. 

WRITTEN     EXERCISES. 

582.  1.  If  exchange  is  at  a  premium  of  -hfo,  and  the 
bank's  charge  is  j^^^,  find  the  cost  of  a  New  York  draft  for 

$300. 

The  exchange  =  i%  of  $300  =  $1.50. 
The  charge  .  =  -,^0^  of  $300  =  .30. 
Total  cost         :=  $300  +  $1.50  +  $.30  =  $301.80. 


414  SCHOOL  ARITHMETIC. 

2.  "When  exchange  was  at  a  discount  of  f^  I  bought  a 
draft  for  $40.  the  bank's  charge  being  10^*.  How  much  did 
the  draft  cost  me  ? 

3.  Find  the  cost  in  New  York  of  a  sight  draft  on  London 
for  £25  8s. 

£25  8s.  =  £25.4. 
25.4   X  14.865  =  $123.57. 

4.  When  New  York  exchange  is  at  ^^  premium,  what 
is  the  cost  of  a  draft  for  $600  ? 

5.  Find  the  cost  of  a  $200  draft  at  30^  a  $1000  discount, 
the  charge  for  issuing  it  being  15^. 

6.  At  50^  a  $1000  premium,  what  is  the  cost  in  St.  Louis 
of  a  draft  on  Boston  for  $540,  if  the  western  bank  charges 
■^^  for  issuing  ? 

7.  When  exchange  was  ^^  discount  I  bought  a  New  York 
draft  for  $1200,  paying  a  local  charge  of  -^^.  Find  the  cost 
of  the  draft. 

8.  When  a  New  York  draft  for  $10000  can  be  bought  in 
Chicago  for  $9800,  is  exchange  at  a  premium  or  at  a  dis- 
count ?  What  is  the  rate  of  exchange  ?  The  bank  of  which 
city,  then,  has  a  large  balance  to  its  credit  in  the  other  city  ? 

9.  AVhat  is  the  cost  of  a  sight  draft,  or  bill  of  exchange,  on 
London  for  £300,  exchange  $4.90  ?  Is  exchange  selling  at  a 
premium  or^t  a  discount  in  this  case  ? 

10.  What  is  the  cost  of  a  sight  draft  on  Paris  for  1000 
francs,  exchange  5.18^  ? 

11.  What  will  be  the  cost  of  a  bill  of  exchange  on  Berlin 
for  1200  marks,  the  rate  of  exchange  being  $.945  for 
4  marks  ? 

12.  If  a  New  York  merchant  owes  $2500  to  a  dealer  in 
London,  and  remits  by  draft,  what  is  the  face  of  the  draft, 
if  the  rate  of  exchange  is  $4.87  ? 

13.  Harry  B.  Naylor,  of  Pittsburg,  draws  on  W.  A. 
Saunders,  of  Toledo,  at  30  days  after  sight  for  $320.  The 
latter  accepts  July  3.     Write  the  draft  and  acceptance. 


AVERAGE  OF   PAYMENTS.  415 


AVERAGE   OF   PAYMENTS. 

583.  1.  For  what  time  is  the  use  of  $1  worth  us  much  as 

the  use  of  $2  for  1  month  ? 

2.  For  what  time  is  the  use  of  110  worth  as  much  as  the 
use  of  $5  for  4  months  ? , 

3.  If  one  half  of  a  debt  is  paid  1  month  before  maturity, 
when  may  the  other  half  be  paid  without  loss  to  either  party  ? 

4.  If  A  uses  $100  of  B's  money  for  2  months,  how  long 
may  B  use  $200  of  A's  money  to  balance  the  favor  ? 

6.  A  owes  B  $100  due  in  2  months,  and  $200  due  in  3 
months.  If  the  first  debt  were  paid  in  1  month,  who  would 
gain  ?  How  much  ?  If  the  payment  of  the  second  debt 
were  deferred  1  month  after  maturity,  who  would  lose  ? 
How  much  ?  Then  at  what  time  may  both  debts  be  paid  by 
a  single  payment  without  gain  or  loss  to  either  party  ? 

WRITTEN     EXERCISES. 

584.  1.  A  owes  B  $300  due  in  3  months,  $400  due  in  4 
months,  and  $500  due  in  7  months.  In  how  many  months 
can  he  pay  the  whole  indebtedness  at  one  time  so  that 
neither  party  shall  lose  ? 

The  use  of  $300  for  3  mo.  =  the  use  for  1  mo.  of  $900. 
The  use  of  $400  for  4  mo.  =  the  use  for  1  mo.  ctf  $1600. 
The  use  of  $500  for  7  mo.  =  the  use  for  1  mo.  of  $3500. 

The  use  of  $1200  for  x  mo.  =  the  use  for  1  mo.  of  $6000. 

.-.x  =  $6000  -I-  $1200,  or  5,  the  number  of  months. 

2.  $800  of  a  debt  is  due  in  6  months,  and  $500  of  it  is  due 
in  8  months.     What  is  the  average  term  of  credit  ? 

3.  Find  the  average  (or  equated)  time  for  the  payment  of 
$2000  due  in  3  mo.,  $1500  due  in  4  mo.,  and  $2500  due  in 
8  mo. 

4.  Find  the  average  time  for  the  payment  of  $300  due  in 
30  days,  $500  due  in  60  days,  and  $200  due  in  90  days. 


416  SCHOOL  ARITHMETIC. 

5.  On  Dec.  1,  1900,  a  merchant  bought  goods  as  follows  : 
1350  on  2  mo.,  1500  on  3  mo.,  1700  on  6  mo.  He  gave  one 
note  in  payment.  At  what  date  should  the  note  be  made 
payable  ? 

6.  Find  the  average  time  for  the  payment  of  $1000  due 
May  31,  $1500  due  June  18,  and  $2000  due  July  9,  reckoning 
the  time  from  May  31. 

7.  $3000  is  due  in  8  months.  If  $1200  is  paid  in  5  months, 
and  $900  in  6  months,  how  long  after  maturity  should  the 
balance  be  paid  ? 

8.  Find  the  average  time  of  payment  on  the  following 
debts  :  Mar.  12,  1901,  $1500  due  in  3  months  ;  Apr.  16,  1901, 
$1000  due  in  2  months ;  May  19,  1901,  $1250  due  in  4 
months. 

The  earliest  date  at  which  any  debt  is  due  is  June  12.     The  $1000  is 
due  4  days  after,  and  the  $1250  is  due  99  days  later. 
The  use  of  $1500  for    0  da.  = 

The  use  of  $1000  for    4  da.  =  the  use  for  1  da.  of  $4000. 
The  use  of  $1250  for  99  da.  =  the  use  for  1  da.  of  |123750. 

.-.  the  use  of  $3750  for  x  da.  =  the  use  for  1  da.  of  $127750. 

.'.x  =  $127750  -^  $3750,  or  34  + . 

June  12,  1901,  +  34  days  =  July  16,  1901. 

9.  Find  the  average  time  of  payment  of  the  following 
debts  :  May  5,  1902,  $1250  due  in  30  days  ;  May  15,  1902, 
$900  due  in  90  days  ;  May  25,  1902,  $1150  due  in  60  days. 

10.  A  man,  Feb.  11,  1900,  gave  a  note  for  $850  payable  in 
4  mo.;  but  he  paid  Mar.  22,  $200;  Apr.  20,  $110;  May  10, 
$150.     When  was  the  balance  due  ? 

CASTING   OUT  NINES. 

585.  Every  power  of  10  is  one  more  than  some  multiple 
of  9. 

Thus,  10  =  9  +  1  ;  10^  =--  11  x  9  +  1  ;  10=^  =  111  x  9  +  1,  etc. 

586.  Every  product  of  a  power  of  10  by  a  number  of  one 


CASTING   OUT   NINES.  417 

digit  is  therefore  some  multiple  of  9,  plus  the  number  repre- 
sented by  that  digit. 

Thus,  40  =  4  X  9  4-  4  ;  500  =  55  x  9  +  5  ;  7000  =  777  x  9  +  7,  etc. 

587.  As  every  number  greater  tlian  0  consists  of  the  sum 
of  such  products,  it  follows  that  every  such  number  is  a 
multiple  of  9,  plus  the  sum  of  the  numbers  represented  by 
its  digits. 

7000  =  777  X  9  +  7 

GOO  =    G6  X  9  +  6 

50  =      5x9  +  5 

__4= 4 

7654  =  848  X  9  +  (7  +  6  +  5  +  4) 
-7654  =  850  nines  4-4.     (7  +  6  +  5  +  4)  =  2  nines  +  4. 

It  is  thus  seen  that  the  excess  of  nines  iy,  any  number  equals  the  excess 
of  nines  in  the  sum  of  the  numbers  represented  by  its  digits.  This  prin- 
ciple may  be  applied  to  test  the  accuracy  of  the  work  in  the  simple  pro- 
cesses of  arithmetic. 

1.  Multiply  857  by  (5S. 

857  ...8  +  5  +  7  =  2  nines  +    2 

68 6  +  8  =  1  nine    -f_5 

6856.  10  =  1  nine    +  l-i 

^^42  equal  excesses. 

58276.  ...5  +  8  +  2  +  7-f6  =3  nines  +  1 J 

The  excess  of  nines  in  the  product  of  the  numbers  equals  the  excess  in 
the  product  of  the  excesses  in  the  factors.  Therefore,  the  work  is  correct, 
unless  it  contains  errors  that  balance,  which  is  quite  improbable. 

2.  Divide  46718  by  263. 

46718  -J-  263  =  177,  with  remainder  167. 
The  excess  of  nines 

In  the  dividend  (46718)  is 8 

In  the  divisor  (263)  is  2 

In  the  quotient  (177)  is .     6 

In  the  product (12)  is 3 

In  the  remainder  (167)  is 5 

In  the  sum (8)  is  . .  8 

27 


418 


SCHOOL  ARITHMETIC. 


Since  the  dividend  equals  the  product  of  the  quotient  and  divisor, 
plus  the  remainder,  the  excess  of  nines  in  the  dividend  =  the  excess  in 
the  sum  of  the  excess  in  the  product  of  the  excesses  of  divisor  and  quo- 
tient, and  the  excess  in  the  remainder.  Therefore,  the  work  may  be 
assumed  to  be  correct. 

3.  Find,  by  casting  out  the  nines,  which  of  the  following 

products  are  incorrect  : 

(a).     7777  X     864  =  6,712,328. 
(b).  67853  X  2976  =  201,930,028. 
(c).     3769  X     235  =  885,715. 

4.  Find,  by  casting  out  the  nines,  which  of  the  following 
quotients  are  correct : 

(a).  1,348,708  -^  498  =  2708,  with  remainder  129. 
(b).  87614  -f-  563  =  155,  with  remainder  349. 
(c).  4000  -T-    23  =    173,  with  remainder  18. 

MEASURES    OF    TEMPERATURE. 


Centigrade. 


Fahrenheit. 


Boiling  point 


of  water. 


Freezing  point 


of  water. 


588.  Temperature  is 
measured  by  an  instru- 
ment called  a  Ther- 
mometer. 

There  are  three  scales 
for  measuring  tempera- 
ture by  means  of  the 
thermometer. 

The  Fahrenheit,    used 
in  this  country  in  ordi- 
nary business,  has  the  freezing  point  of  water  marked  32°, 
and  the  boiling  point  212°. 

The  Centigrade^  generally  used  in  science,  has  the  freezing 
point   0°,  and  the  boiling  point  100°. 

The  Reaumur,  which  is  also  frequently  used,  has  the  freez- 
ing point  0°,  and  the  boiling  point  80°. 

Degrees  below  0°  are  indicated  by  the  sign  — .     Thus, — 20  means  20° 
below  zero. 


SPECIFIC  GRAVITY.  419 

WRITTEN   EXERCISES. 

689.  1.  80°  Fahrenheit  corresponds  to  what  temperature 
Centigrade  ? 

212°  F.  -  32°  F.,  or  180°  F.  =  100°  C. 
1°  F.  =  \U°  C.  =  §°  C. 
80°  F.  =  80°  F.  -  32°  F.,  or  48°  F.  above  freezing. 
48  X  i°  =  26.67°. 
That  is,  80°  F.  =  26.67°  C. 

2.  60°  C.  corresponds  to  what  temperature  F.  ? 

100°  C.  =  180°  F. 

60°  C.  =  1%%  of  180°  F.,  or  108°  F.  above  freezing. 
That  is,  60°  C.  =  108°  F.  +  32°  F.,  or  140°  F. 

3.  80°  C.  corresponds  to  what  temperature  R.  ? 

100°  C.  =  80°  R. 
80°  C.  =  AH)  of  80°  R.,  or  64°  R. 

4.  Express  30°  F.  in  Centigrade  scale  ;  in  Reaumur's  scale. 
6.  Express  —35°  F.  in  Centigrade  scale;  in  Reaumur's  scale. 

6.  Express  —  40°  C.  in  Fahrenheit's  scale  ;  in  Reaumur's 
scale. 

7.  Express  —  33°  C.  in  Fahrenheit's  scale  ;  in  Reaumur's 
scale. 

8.  The  temperature  of  a  room  is  63°  F.  Find  the  tem- 
perature in  C.     In  R. 

9.  Express  in  Centigrade  scale  the  following  melting 
points  :  (a)  lead,  630°  F.;  (b)  ice,  32°  F.;  (c)  silver,  873°  F.; 
(d)  tin,  455°  F. 

10.  Express  in  Fahrenheit  scale  the  following  boiling 
points  :  (a)  alcohol,  78°  C;  (b)  ether,  35°  C;  (c)  mercury, 
357°  C. 

SPECIFIC    GRAVITY. 

690.  The  Specific  Gravity  of  any  substance  is  the  ratio 
of  its  weight  to  the  weight  of  an  equal  bulk  of  water. 

Thus,  a  cubic  foot  of  zinc  weighs  7000  oz.,  and  a  cubic  foot  of  water 
1000  oz.  The  ratio  of  7000  oz.  to  1000  oz.  is  7  ;  hence  the  specific  gravity 
of  zinc  is  7.    That  is,  zinc  is  7  times  as  heavy  as  water. 


420  SCHOOL  ARITHMETIC. 

Table  of  Specific  Gravity. 

Copper 8.9  Nickel 8.9  Cork.......     .24 

Gold 19.3  Silver 10.5  Granite.. ..  2.7 

Lead 11.3  Sulphur 2.0  Steel 7.8- 

Alcohol 79  Petroleum .7  Mercury. .  .13.596 

591.  If  a  substance  is  in  water,  the  water  buoys  it  up  by 
just  the  weight  of  the  water  displaced  by  it.  That  is,  it 
loses  a  portion  of  its  Aveight  just  equal  to  the  weight  of  the 
water  displaced. 

WRITTEN    EXERCISES. 

592.  1.  If  a  cubic  foot  of  iron  Aveighs  487.5  lb.,  and  an 
equal  volume  of  water  weighs  62.5  lb.,  what  is  the  specific 
gravity  of  the  iron  ? 

487.5  lb.  -i-  62.5  lb.  =  7.8,  the  specific  gravity. 

.    2.  What  is  the  weight  of  a  cubic  inch  of  silver  ? 

1  cubic  foot  of  water  weighs 1000  oz. 

1  cubic  inch  of  water  weighs |9f  ^  oz. 

,*.  1  cubic  inch  of  silver  weighs 10,5  x  |^|*^  oz.  =6+  oz, 

3.  If  a  body  weighs  3.71  Kg.  in  air  and  2.38  Kg,  in  water, 
what  is  its  specific  gravity  ? 

3.71  Kg.  -  2.38  Kg.  =  1.33  Kg.     Since  the  body  weighs  1.33  Kg.  less 
in  water  than  in  air,  1.33  Kg.  is  the  weight  of  the  water  displaced  by  it. 
3.71  Kg.  -^  1.33  Kg,  =  2.8  nearly,  the  specific  gravity  of  the  body. 

4.  What  does  a  bar  of  aluminum  113  mm.  long,  17  mm. 
wide,  and  13  mm.  tliick  weigh  if  its  specific  gravity  is  2.57  ? 

5.  If  a  bar  of  iron  18  in.  long,  2^  in.  wide,  If  in.  thick, 
weighs  18  lb.  9  oz.,  what  is  the  specific  gravity  of  the  iron  ? 

6.  How  many  pounds  does  a  man  lift  in  raising  a  cubic 
foot  of  stone  under  water  if  its  specific  gravity  is  2.5  ? 

7.  If  the  specific  gravity  of  gold  is  19.3,  find  the  number 
of  cubic  inches  of  gold  to  the  pound. 

8.  How  many  cubic  feet  of  sea  water  weigh  a  ton,  if  its 
specific  gravity  is  1.026  ? 

9.  The  specific  gravity  of  ice  is  .92,  of  sea  water  J.. 025. 
To  what  depth  will  a  cubic  foot  of  ice  sink  in  sea  water  ? 


INTRODUCTION    TO    ALGEBRA. 

593.  In  passing  from  arithmetic  to  algebra  the  meaning 
of  number  and  the  method  of  representing  it  are  extended, 
but  there  is  nothing  contradictory  to  what  has  been  already 
learned  in  arithmetic.  Algebra  may  be  regarded  as  but  a 
more  comprehensive  arithmetic. 

The  symbols,  1,  2,  3,  etc.,  are  retained  in  algebra  with  their  arithmetic- 
al meanings,  and  the  same  symbols,  +,  — ,  x,  -*-,  (  ),  =,  are  used  in 
each.  Fractions,  powers,  and  roots  have  the  same  meaning  and.  are 
written  in  the  same  form. 

Literal  or  General  Number. 

594.  An  important  difference  between  arithmetic  and 
algebra  comes  from  the  frequent  and  extended  use  in  the 
latter  of  letters  to  represent  numbers.  Just  as,  in  interest 
problems,  p  may  stand  for  principal,  r  for  rate  per  cent,  t  for 
the  time,  i  for  interest,  and  a  for  amount,  so  in  any  case  such 
symbols  as  a,  b,  x,  y  may  be  used  to  represent  any  numbers 
whatever. 

In  arithmetic  we  speakof  5  books,  meaning  a  certain  number  of  books, 
or  of  $10,  meaning  a  certain  number  of  dollars  ;  in  algebra  we  speak  of 
71  books,  meaning  any  number  or  an  unknoivn  number  of  books,  of  x  dol- 
lars, meaning  any  number  or  an  unknown  number  of  dollars, 

595.  Numbers  represented  by  letters  are  called  Literal  or 
General  Numbers.  The  reasoning  is  the  same  whether  num- 
bers are  represented  by  letters  or  by  figures. 

Thus,  if  a  stands  for  a  certain  number,  say  the  number  of  pupils  in  a 
room,  then  2a  stands  for  twice  this  number,  3a  for  three  times  this  num- 
ber, etc. 


422  SCHOOL  ARITHMETIC. 

1.  If  n  stands  for  the  number  of  books  in  my  library,  what 
is  the  meaning  of  3w  ?     Of  bn ?     OiQn?     Of  ^n  ? 

2.  If  X,  y,  and  z  stand  for  the  cost  of  a  horse,  a  cow,  and  a 
sheep  respectively,  for  what  does  x  +  y  stand  ?  x  +  y  +  z? 
2x  +  ^  +  z? 

3.  If  a:  =  5,  ?/  =  9,  and  2;  —  7,  what  is  the  value  of  x  -\-  y? 
Otx  +  y  —  z? 

4.  If  in  a  number  of  two  digits,  the  digit  in  the  ones' 
place  is  5,  and  the  digit  in  the  tens'  place  is  2,  the  number  is 
10  X  2  +  5.     Write  a  number  containing  a  ones  and  b  tens. 

Positive  and  Negative  Numbers. 

596.  Sometimes  quantities  that  are  measured  by  the  same 
unit  are  of  opposite  qualities.  Thus,  assets  and  liabilities  are 
both  measured  by  the  unit  dollar,  the  readings  of  a  ther- 
mometer above  and  below  zero  are  given  in  degrees,  and 
dates  A.D.  and  dates  B.C.  are  both  given  in  years. 

In  the  case  of  assets  and  liabilities,  the  unit  dollar  may  be  taken  either 
as  a  dollar  of  assets  or  as  a  dollar  of  liabilities  ;  if  as  a  dollar  of  assets, 
then  assets  are  regarded  as  positive  ;  and  liabilities,  for  the  sake  of  dis- 
tinction, as  negative.  In  order  to  represent  quantities  that  have  opposite 
qualities  we  need  to  extend  the  idea  of  number  as  given  in  arithmetic  so 
as  to  include  numbers  that  count  negative  units. 

697.  The  numbers  arithmetic  deals  with  are  greater  than 
zero,  and  are  called  positive  numbers  ;  but  algebra  treats  also 
of  numbers  that  in  relation  to  positive  numbers  are  regarded 
as  less  than  zero,  and  these  are  known  as  negative  numbers. 

698.  The  primary  notion  of  a  negative  number  is  that  of 
one  which,  when  taken  with  a  positive  number  of  the  same 
kind,  goes  to  diminish  it,  cancel  it,  or  reverse  it. 

Thus,  liabilities  neutralize  so  much  assets,  thereby  diminishing  the 
net  assets,  canceling  them,  or  leaving  a  net  liability. 

699.  Negative  numbers  may  be  regarded  as  arising 
through  the  extension  of  the  operation  of  subtraction  to  the 


INTRODUCTION  TO   ALGEBRA.  423 

case  ill  which  the  minuend  is  less  than  the  subtrahend, 
which,  from  an  arithmetical  point  of  view,  is  impossible. 
Note  the  following  : 

6-4  =  2. 

5-4  =  1. 

4-4  =  0. 

3  —  4  =  - 1  ;  that  is,  a  numher  one  unit  less  than  0. 

2  —  4  =  -2  ;  that  is,  a  number  two  units  less  than  0. 
Observe  that,  as  the  minuend  decreases  by  1,  3,  or  more  units,  the 
subtrahend  remaining  the  same,  the  remainder  decreases  by  an  equal 
number  of  units,  becoming  0  when  the  minuend  is  equal  to  the  subtra- 
hend. If,  then,  the  minuend  becomes  less  than  the  subtrahend  by  1,  2, 
or  more  units^the  remainder  must  decrease  by  an  equal  number  of  units, 
and  therefore  become  less  than  0  by  1,  2,  or  more  units.  But  the  opera- 
tion of  subtracting  a  greater  number  from  a  less  is  possible  only  when 
numbers  less  than  zero  are  introduced. 

600.  The  negative  remainder,  -1,  does  not  mean  that 
more  units  were  taken  from  the  minuend  3  than  it  con- 
tained ;  it  merely  shows  that  the  subtrahend  is  1  unit  greater 
than  the  minuend. 

601.  The  absolute  value  of  a  number  is  the  number  of 
units  contained  in  it  without  regard  to  their  quality.  The 
numbers  of  arithmetic  are  the  absolute  values  of  the  positive 
and  negative  numbers  of  algebra.  Since  letters  are  used  to 
represent  numbers  which  may  have  any  values  whatever, 
they  can  represent  either  positive  or  negative  numbers. 

602.  In  the  expression  6  —  4,  the  minus  sign  indicates 
that  4  is  to  be  taken  from  6.  It  is  a  symbol  of  operation, 
and  does  not  show  4  to  be  a  negative  number.  Both  6  and 
4  are  positive  numbers. 

But  the  same  sign  has  another  use,  namely,  to  denote 
negative  numbers ;  and  the  sign  +  is  used  to  denote  positive 
numbers.  When  so  used  these  signs  are  symbols  of  quality, 
and  do  not  indicate  any  operation  whatever. 

Thus,  -4  means  a  number  four  units  less  than  0,  and  +4  a  number  four 
units  greater  than  0. 


424 


SCHOOL  ARITHMETIC. 


603.  Positive  and  negative  numbers  are  called  opposite 
numbers,  and  may  represent  any  quantities  that  are  opposite 
in  their  relation  to  each  other. 

Thus,  degrees  ahoi^e  zero  on  a  thermometer  may  be  called  positive  ; 
degrees  below  zero,  negative ;  distance  east,  positive  ;  distance  west, 
negative  ;  assets,  positive  ;  liabilities,  negative. 

604.  Zero  is  neither  a  positive  nor  a  negative  number  ;  it 
is  the  starting  point  from  wliich  positive  and  negative  num- 
bers are  counted. 

Thus,  opposite  temperatures  are  counted  from  zero  on  the  ther- 
mometer, —5°  meaning  5  degrees  helow  zero,  and  +5°  meaning  5  degrees 
above  zero. 

605.  Cash  received  and  cash  spent  are  opposite  quantities, 
and  may  be  represented  by  positive  and  negative  numbers. 

Thus,  $50  received  may  be  represented  by  +  $50. 

by  -$50. 
A  man's  cash  account  might 
be  kept  as  in  the  left-hand 
column,  or  as  in  the  right. 
Friday  he  looks  over  his  fig- 
ures to  see  how  much  cash  he 
ought  to  have  on  hand.  He 
adds  the  sums  received,  which 
amount  to  $44,  and  the  sums 
expended,  which  amount  to 
$44.  These  cancel  each  other. 
That  is,  $44  received  united  with  $44  spent  is  equal  to  neither  cash  on 
hand  nor  debt  ;  or 

$44  received  +  $44  spent  -—  0. 

Or,  he  may  add  the  positive  numbers  in  the  column  to  the  right,  get- 
ting +  $44,  and  the  negative  numbers,  getting  -$44.  This  may  be  ex- 
pressed algebraically  thus  : 

+  $44  +  -$44  =  0. 

606.  We  have  already  seen  that  zero  is  the  difference  be- 
tween two  equal  numbers.  From  the  preceding  article  we 
learn  that  it  is  also  the  sum  of  two  equal  and  opposite  num- 
bers. 


$50  spent 

<(     (( 

Monday, 

received    $25, 

or  +125 

(( 

spent         $  8, 

or  -$  8 

Tuesday, 

spent         $13, 

or  -$13 

Wednesday, 

received    $  9, 

or  +$  9 

<( 

spent         116, 

or  -$1G 

Thursday 

$  7, 

or  -$  7 

(' 

received     $10, 

or  +$10 

Friday,  Cash  on  hand,       0 

0 

INTRODUCTION  TO  ALGEBRA.  425 

Thus,  gain  $5  +  loss  $5  =  0. 
Or,  in  algebraic  language,  +$5  +  -|5  =  0. 

Carefully  examine  the  following  statements  : 
20  dollars  gain  +  20  dollars  loss  =  0. 
+^0  +  -20  =  0. 
This  result  means  neither  ffain  nor  loss. 

20  dollars  gain  +  15  dollars  loss  =  5  dollars  gain. 
+20  +  -15  =  +5. 
This  result  means  a  net  gain  of  5  dollars. 

20  dollars  gain  +  30  dollars  loss  =^  10  dollars  loss. 
+  20  +  -30  =  -10. 
This  result  means  a  net  loss  of  10  dollars. 

5  miles  east  +  5  miles  west  =  the  starting  point. 
+  5  +  -5  =  0. 
This  result  means  that  the  traveler  has  returned  to  the 
point  from  which  he  started. 

10  miles  noftJi  +  15  miles  south  =  5  miles  south. 
+  10  +  -15  =  -5. 
This  result  means  that  the  traveler  stopped  5  miles  south 

of  his  starting  point. 

Make  similar  statements  for  each  of  the  following  : 

1.  180  gain,  |50  loss. 

2.  175  gain,  $100  loss. 

3.  40  miles  east,  30  miles  west. 

4.  A  rise  of  20°  in  tem'perature,  then  a  fall  of  18°. 

6.  An  army  balloon  ascended  3000  feet,  then  fell  1800  feet. 

ADDITION. 

607.  In  algebra  the  process  of  adding  two  or  more  pos- 
itive or  negative  numbers  is  the  same  as  that  of  adding  in 
arithmetic,  except  that  the  sigti  of  quality  is  to  be  prefixed 
to  the  sum. 

Thus,  +7  added  to  +3  =  +10  ;  -7  added  to  -3  =  -10. 


426  SCHOOL  ARITHMETIC. 

Add  the  following  : 

1.  +9to+l().      4.   +8.4  to +9.9.  7.  ^8{a  +  b)  to +3{a  -^  b). 

2.  +84  to  +48.     5.   +8yto+e)y.  8.  -7{m  —  n)  to-6{m  ■- n). 

3.  -72  to  -28.    6.  -Via  to  -oa.  9.  +5a;  and  +x  to  +9a;. 

608.  Terms  containing  the  same  letters  with  the  same 
exponents  are  called  Similar  Terms. 

Thus,  dy^  and  -5y^  are  similar  terms,  as  are  7  {x  +  y)  and  2  (x  +  y). 

600.  Principle. — As  in  arithmetic  only  like  numbers 
can  be  added,  so  in  algebra  only  similar  algebraic  7iumbers 
can  be  united  by  addition  into  one  term. 

Although  unlike  numbers  can  not  be  united  by  addition  into  one  term, 
an  indicated  operation  is  regarded  as  their  algebraic  sum.  Thus,  m  +  n 
is  called  the  sum  of  m  and  n. 


Add  the  following : 

1.                  2. 

3. 

4. 

5. 

+^x                 -X 

5ab 

"^my 

(a  +  b) 

^x               -dx 

eab 

-'Hmy 

b(a  +  b) 

+9a:                -9x 

ab 

-7  my 

4(«  -\-  b) 

When  numbers  are  positive,  the  symbol  of  quality  (  +  )  is  usually 
omitted,  as  in  examples  3  and  5  above.  When  no  symbol  is  written,  + 
is  understood.     The  sign  —  is  never  omitted. 

610.  The  following  equations  were  considered  in  article 
606: 

+20  +  -20  =     0         •         (1). 
+20  +  -15  =  +5  (2). 

+20  +  -30  ^  -10  (3). 

In  (1)  we  see  that  +20  and  -20  cancel  each  other,  that  is, 
that  their  sum  is  0. 

In  (2)  we  see  that  -15  cancels  +15,  leaving  5  positive 
units  (+5),  which  is  the  sum. 

In  (3)  we  observe  that  +20  cancels  -20,  leaving  10  nega- 
tive units  (-10),  which  is  the  sum. 

Queries.— How  is  the  5  in  (2)  obtained  ?    Why  has  it  the  +  sign  ?    In 


INTRODUCTION  TO  ALGEBRA.  427 

(3)  how  is  the  10  obtained  ?    Why  has  it  the  —  sign  ?    Is  -2x  the  sum 
of  5x  and  -7x  ?    Why  ? 

Add  the  following : 

1.  +8  to  -20.     4.  +3x  to  -12a;.       7.  Sx'y  to  -llx'y. 

2.  -T  to  +15.     6.  ISab  to  -5«^.      8.  8(m  +  ?i)  to  -7{m  +  n). 

3.  25  to  -16.     6.  -2%  to  +30^.     9.  -{x  -  y)  to  10(a;  —  y). 
10.  Find  the  algebraic  sum  of  bx,  -7a:,  +9a;,  -4a;,  and  a;. 

1.  The  sum  of  the  positive  numbers  is  15a: ;  the  sum  of  the  negative 
numbers  is  -11a;.     These  two  sums  united  =  Ax. 

2.  In  this  example  the  5,  7,  9,  and  4  are  coefficients.     When  no  coef- 
ficient is  expressed,  1  is  understood. 

Add  the  following  : 

11.                12.  13.  14. 

^ax            -97nn  SUcd  72  {b  -  a) 

-5ax           -mn  -2obcd  -48  (b  —  a) 

+  7ax            +4mw  -9bcd  -50  (b  -  a) 

Express  in  the  simplest  form  : 

15.  8a;  4-  3a:  —  5a:  +  a;   -  4a;  +  12a;  —  7a;. 

16.  15Z»a:  —  6bx  -{-  bx  —  9bx  —  bx  +  18bx  —  lObx. 

17.  5(m  —  7i)  +  13(m  —  7i)  —  ll(m  -  n)  +  Q(m-n)  — 
20(m  —  ii). 

18.  Add  3a:    +    «  —   2y,  5a;  —  4«    +    6z/,  7«  —  8y,  and 
y  —  4a;  +  6a. 

3a:  +      a  —  2y 

bx  —     4ca  +  6y  For  convenience  we  write  similar  terms  in  the 

^a  —  8y  same  column.     The  sum  of  the  first  column  is 

-4a:  +     6a  -^     y  +4x,  of  the  second  +10a,  and  of  the  third  -3y. 

4a:  -f-  10a  —  3y 

19.  Add  6y  —  4c,  3?/  +  8c,  5c  —  4?/,  y  —  c,  y  —  10c. 

20.  Add    6m  +  2?^  —  bb,    7?i  —  3&  —  4???,    b  +  8m  —  9n, 
m  +  bb. 

21.  If  a  man  has  a  sons,  ^  daughters,  and  1  wife,  how 
many  persons  are  in  the  family  ? 


428  SCHOOL  ARITHMETIC. 

22.  A  boy  who  had  15  cents  found  in  cents  and  earned  4m 
cents.     How  much  had  he  then  ? 

23.  My  house  is  d  feet  long  and  c  feet  wide.  What  is  the 
distance  around  it  ? 

24.  Tom  walked  due  east  m  hours,  then  due  west  n  hours. 
If  his  rate  was  3  miles  an  hour,  at  what  distance  from  his 
starting  point  did  he  stop  ? 

25.  If  m  in  the  above  problem  is  equal  to  7i,  where  did 
Tom  stop  ? 

26.  If  m  =  5  and  n  =  d,  how  far,  and  in  what  direction, 
from  his  starting  point  did  he  stop  ? 

27.  Locate  his  stopping  place  if  m  =  4  and  n  =  Q. 

28.  D  earns  a  dollars  each  week  and  spends  b  dollars. 
How  much  will  he  have  at  the  end  of  8  weeks  ? 

29.  What  will  be  his  financial  condition  it  a  =  $15  and 
d  =  UO?     What  will  it  be  if  «  =  $12  and  ^>  =  $16  ? 


SUBTRACTION. 

611.  In  algebra,  as  in  arithmetic,  the  minuend  is  the  sum 
of  the  subtrahend  and  the  remainder. 

Thus,   10  +  -4  =  0,  the  sum,  which  we  may  regard  as  a  minuend. 
Taking  10  as  the  subtrahend,  we  have 

6  —  10  =  -4,  the  remainder. 
Taking  -4  as  the  subtrahend,  we  have 

6  —  -4  =  10,  the  remainder. 
In  either  case  6  is  the  sum  of  10  and  -4. 

Carefully  examine  the  following: 

8-2  =  6.  -11  -       2  =  -13.         1  -     2  =  -1. 

8  +  -2  =  6.  -11  +     -2  =  -13.         1  +  -2  r^  -1. 

8=6  +  2.  -11  =  -13  +     +2.         1  =  -1  +     2. 

Observe  l}\&t  subtracting  +2  is  equivalent  to  adding  -2  ;  also,  that  the 
minuend  is  the  sum  of  the  remainder  and  the  subtrahend. 


INTRODUCTION   TO   ALGEBRA.  429 

Examine  these  equations  : 

7  -    -4  =  11.         -12  -  -4  =  -8.  2  -  -4  =     6. 

7  +       4  =  11..      -12  +     4  =  -8.  2  +     4  =     6. 

7  =     11  +  -4.         -12  =  -8  +  -4.  2=6  +  -4. 

Observe  that  subtracting  -4  is  equivalent  to  adding  +4  ;.also,  that  the 
minuend  =  remainder  +  subtrahend. 

012.  In  general,  to  subtract  n positive  number  is  equivalent 
to  adding  an  equal  negative  number  ;  to  subtract  a  negative 
number  is  equivalent  to  adding  an  equal  positive  number. 

Illustration  I. 

A  man  wliose  income  is  $100  a  month  spends  $00,  and 
saves  $40.  If  his  income  is  reduced  $10  a  month,  he  will 
save  $30.     $90  -  $60  =  $30. 

Or,  if  his  expenses  are  increased  $10  a  month,  he  will  save 
$30. 

$100  -  $70  =  $30. 

Hence,  to  tahe  away  $10  income  is  equivalent  to  adding 
$10  expenses.     Either  reduces  his  savings  to  $30. 

Calling  income  and  ^ixsmg^ positive,  and  expenses  ^e^a^eve, 
we  have  the  following  algebraic  expression  of  this  relation  : 
$40  -  +$10  =  $30. 
$40  +  -$10  =  $30. 

Illustration  II. 
If  his  income  is  increased  $10  a  month,  he  will  save  $50. 

$110  -  $60  =  $50. 
Or,  if  his  expenses  are  reduced  $10  a  month,  he  will  save 
$50. 

$100  -  $50  =  $50. 

Hence,  to  tahe  away  $10'  expenses  is  equivalent  to  adding 
$10  income.  Either  increases  his  savings  to  $50.  The  rela- 
tion is  algebraically  expressed  thus  : 

$40  -  -$10  =  $50. 

$40  +   +$10  =  $50. 


430  SCHOOL  ARITHMETIC. 

1.  A  has  $10 ;  B  has  no  money  and  is  $5  in  debt.  How 
much  more  is  A  worth  than  B  ? 

2.  Tom  has  $50  in  bank.  Harry  has  $20  in  cash,  but  owes 
John  140.     Tom  is  worth  how  much  more  than  Harry  ? 

3.  Albert  has  24  marbles  and  Ed  has  none,  but  owes  Albert 
16.     How  many  more  marbles  than  Ed  has  Albert  ? 

4.  Mary  has  8  jacks  and  Alice  has  -5  (i.e.,  owes  5).  Alice 
has  how  many  fewer  than  Mary  ? 

5.  In  the  schoolroom  the  temperature  is  70°  above  zero, 
while  outside  it  is  5°  below  zero.  How  many  degrees  warmer 
is  it  inside  tlian  outside  ?     75°  —  -5°  =  (     ). 

6.  A  is  $35  in  debt  and  B  is  $50  in  debt.  How  much  bet- 
ter off  is  A  than  B  ?     -$35  -  -$50    =  (     ). 

613.  In  algebra  when  the  subtrahend  is  greater  than  the 
minuend,  the  remainder  is  negative,  and  shows  liotv  nmchiYiQ 
subtrahend  exceeds  the  minuend. 

Thus,  8  -  11  =  -3  ;  5a:  -  9a:  =  --4a;  ;  -\2y  -  -5y  =  -ly. 

Subtract  the  following : 

1.  8  from  5.  5.-1  from  -11.  9.     15«  —  22a. 

2.  9  from  1.  6.     -8  from  -5.  10.     -4y  from  -lly. 

3.  25  from  7.  7.   -27  from  -2.  11.     -9a:  from  -20:r. 

4.  -6  from  -10.       8.  7a:  -  15.r.  12.   -2U  from  -18^. 

614.  In  arithmetic  the  remainder  is  never  greater  than 
the  minuend  ;  but  in  algebra  it  is  often  greater.  Note  the 
following  : 

(a)  (b)  (c)  (d) 

Minuend +31  -19  +12  -35 

Subtrahend  . .  +16  +13  ^  -24 

Remainder       +15  -32  +20  -1 

In  (a)  and  (b)  the  remainders  are  less  than  the  minuends  ;  in  (c)  and 
(d)  they  are  greater.     Which  subtrahends  are  positive  ? 

These  examples  illustrate  the  following  : 

1.  When  the  subtrahend  is  j^ositive  the  remainder  is  less 


INTRODUCTION  TO   ALGEBRA.  431 

than  the  minuend.  Subtracting  a  positive  number  is  equiva- 
lent to  adding  an  equal  negative  number. 

2.  When  the  subtrahend  is  negative  the  remainder  is 
greater  than  the  minuend.  Subtracting  a  negative  number 
is  equivalent  to  adding  an  equal  positive  number. 

1.  From  12a;  subtract  bx. 
12x         12a;  —  +bx  =  Ix  (subtracting  a  positive  number); 

hx  or  12a;  +  ~5a;  =  7a;  (adding  a  negative  number). 
~x 


\.  Subtract  : 

I8y  from  Iby. 

15y 

15y- 

-  +18y 

=  -3y. 

18y 

i5y  ■ 

f  -18y 

=  -3y. 

-3y 

1.  From  7a  take  -5a 

',. 

7« 

7«  - 

-5a  = 

12a. 

-5a 

7a  + 

5a  = 

12a. 

12a 

\,  From  4a; 

—  3y  take  x  + 

^y^ 

4a;  -  Zy 

4a;  - 

-  X  = 

Zx,  and 

-%y- 

+  2^  = 

-oy, 

x  +  2y 

or   - 

-3y  4- 

-%y  = 

-5y. 

3x  —  by 

Notice  that  in  each  of  these  examples  we 

have  changed  the 

8ign  of  the 

subtrahend  and  then  added. 
Subtract : 
6.  8a;  from  3a;.  9.  7a;  —  2«/  from  8a;  —  5y. 

6.  13?/  from  -17^.  10.  9«J  +  '6d  from  12«^'  -  d. 

7.  -5a  from  -12a.  11.  a  —  h  from  a  +  ^. 

8.  -Qh  from  4^.  12.  a;  —  «/  +  0  from  x  +  y  -\-  z. 

13.  3aa;  —  7^  +  55  from  6aa;  —  4i/  —  3J. 

14.  4c(5  —  ^)  +  6am  from  10c(J  —  </)  +  6am. 

15.  Find  the  value  of  5a  +  7^*  -  (3a  +  U). 

Both  da  and  3&  are  to  be  subtracted  from  the  minuend.     Hence,  when 


432  SCHOOL  ARITHMETIC. 

the  parenthesis  is  removed,  tlie  signs  of  both  must  be  changed  to  —  > 
and  we  have  5a  +  7b  —  Za  —  2b,  or  2a  +  56. 

16.  12.C  +  Sy  -  (dx  -  y)  =  what  ? 

I2x  +  ^y  -  ^x  +  ij  =  ^x  +  9y. 
Complete  the  following  equations  : 

17.  3^  +  17:?;  -  {y  +  16a;)  = 

18.  5(m  -\-  n)  —  Ixy  —  (m  4-  w)  +  4:xy  = 

19.  12(a  -{-  b  +  c)  -  8  {a  +  b  -h  c)  ~  U  -h  20g  = 

20.  By  selling  a  cow  for  a  dollars  I  gained  b  dollars. 
What  did  the  cow  cost  ? 

21.  A  and  B  together  have  10  children,  of  whom  A  has  n. 
How  many  has  B  ? 

22.  Smith  has  b  dollars  and  Jones  is  c  dollars  in  debt. 
How  much  more  is  Smith  worth  than  Jones  ? 

23.  H'  I  have  8  children  and  m  of  them  are  girls,  how 
many  boys  have  I  ? 

24.  A  man  who  earns  d  dollars  a  month  spends  p  dollars 
for  rent  and  q  dollars  for  other  purposes.  How  much  does 
he  save  in  a  month  ? 

25.  A  boy  wishing  to  ride  n  miles  on  his  wheel  rode  a 
miles  the  first  day  and  b  miles  on  each  of  the  next  two  days. 
How  many  miles  had  he  yet  to  ride  ? 

26.  A  man  engaged  to  fence  a  field  c  rods  long  and  d  rods 
wide.  If  he  built  (c  —  d)  rods  a  day  for  3  days,  how  many 
rods  had  he  yet  to  build  ? 

27.  A  drover  paid  |m  for  b  pigs,  and  $wfor  c  sheep.  He 
sold  the  former  at  $8  and  the  latter  at  $5  a  head.  Find  his 
gain. 

MULTIPLICATION. 

615.  When  the  multiplier  ispositive,  the  process  of  multi- 
plication in  algebra  is  the  same  as  that  in  arithmetic,  except 
that  the  sign  of  quality  of  the  multiplicand  is  to  be  written 
before  the  product.     Thus, 


INTRODUCTION  TO  ALGEBRA.  .         433 

(a).  +3  X  +5  =  +15. 
(b).  +3  X  -5  =  -15. 

Observe  that  three  times  5  positive  units  gives  15  positive  units  as  a 
iroduct,  and  three  times  5  negative  units  gives  15  negative  units  as  a 

iroduct. 

616.  When  the  multiplier  is  negative^  it  gives  to  the 
)rodiict  a  sign  opposite  to  that  given  by  a  positive  multiplier; 
hat  is,  the  quality  of  the  multiplicand  is  reversed  in  the 
)roduct.     Note  the  following  :• 

(c).     -3  X   +5  =  -15. 
(d).     -3  X  -5  =  +15. 

Since  +3  +  -3  =  0,  -3  =  0  -  +3,  or  -(+3)  ;  hence  in  (c),  -3  x  +5 
lay  be  regarded  as  signifying  that  +5  is  to  be  taken  three  times,  and 
hen  tlie  result  reversed  (subtracted),  giving  the  product  the  sign  opposite 
0  that  of  the  multiplicand. 

Similarly  in  (d),  -3  x  -5  may  be  regarded  as  signifying  that  -5  is  to 
e  taken  three  times,  and  the  result  reversed,  that  is,  -15  is  to  be  sub- 
racted.  Now,  subtracting  -15  is  equivalent  to  adding  +15.  .'.  -3  x 
5  =  +15. 

Queries. — 1.  In  (a)  and  (b)  the  product  has  the  sign  of  the  raultipli- 
and.     Is  this  true  in  (c)  and  (d)  where  the  multiplier  is  negative? 

2.  When  both  factors  are  positive,  as  in  (a),  what  is  the  sign  of  the 
iroduct  ?  When  both  are  negative,  as  in  (d)  ?  When  they  have  unlike 
igns,  as  in  (b)  and  (c)  ? 

617.  If  a  and  h  stand  for  any  two  numbers,  we  have 

+a  X   +h  =  -^ah, 

+a  X  -b  =  -ah, 

-ax   +b  =  -ab, 

^  -a  X  -b  =  +ab. 

That  is,  when  two  factors  have  like  signs  the  product  is 

wsitivej  when  they  have  unlike  signs  the  product  is  negative. 

Illustrations. 

A  train  whose  speed  is  20  miles  an  hour  runs  north  and 
outh  past  a  point  P,  passing  at  12  m.     Locate  the  train  at  5 
'.M.  and  at  7  a.m. 
28 


iBi  SCHOOL  ARITHMETIC. 

Consider  the  following  as  positive :  (1)  distances  north, 
(2)  the  train's  rate  northward,  (3)  time  after  12  m.  Consider 
the  opposites  negative.  Then,  if  the  train  is  running  north- 
ward, 

(a)  in  5  hours  after  12  M.  it  will  be  100  miles  north  of  P, 
which  is  expressed  algebraically  by 

+  5   X  +20  =  +100. 

(b)  5  hours  before  12  M.  it  will  be  100  miles  south  of  P, 
expressed  algebraically  by 

-5  X   +20  =  -100. 
If  the  train  is  running  southward, 

(c)  in  5  hours  after  12  M.  it  will  be  100  miles  south  of  P, 
expressed  algebraically  by 

+  5  X  -20  =  -100. 

(d)  5  hours  before  12  m.  it  will  be  100  miles  north  of  P, 
expressed  algebraically  by 

-5  X  -20  =  +100. 

Multiply  the  following  : 

1.  4a     by  3.  6.  9xy  by  -a. 

2.  -b     by  7.  7.  -ab  by  10. 

3.  -6x  by  2.  8.  -c'd  by  -5. 

4.  -6y  by  -d.  9.  -12   by  ax. 

5.  -8m  by -1.  10.  -1     by  15/?. 

11.  What  is  the  product  of  a  —  b  +  2x  multiplied  by  Sa  ? 

^        ^    '    '^'^  Each  terra  of  the  multiplicand   is  multiplied 

^a  by  3a.     The  algebraic  sum  of  these  products  is 

3«'  —  'dab  +  6ax      "^^^  required  product. 

Multiplication  is  usually  indicated  by  writing  letters,  or  a  figure  and 
one  or  more  letters,  side  by  side. 


INTRODUCTION  TO  ALGEBRA. 


485 


12.  Find  the  product  of  x^  +  ^x^y  —  Zxy^  — y"  multiplied 
by  ic  -  y. 

x^  +  '^x^y  -  2>xy''  —  y^ 
X  —  y 

X*  +  ^x^y  —  Sx^y^—  xy^  =  product  of  multiplicand  by  x. 
—  x^y  —  3a;y  +  3xy*  +  y*  =  product  of     ^^  by  -y. 


X*  +  2x^y—  6a;y  +  2xy^  +  y*  =  complete  product. 

It  is  a  convenient  arrangement  to  write  the  multiplier  under  the 
multiplicand,  and  place  like  terms  of  the  partial  products  in  columns. 
Observe  that  in  multiplying  we  take  the  product  of  the  coeflBcients  and 
the  sum  of  the  exponents  of  the  same  letters. 


19.  8«m  +  Qbn  by  bah  —  h. 

20.  a;'  +  2xy  +  y^  hy  x  +  y. 

21.  {i?i  —  ny  by  (m  —  n). 

22.  a'-h  ab  +  hc-  V  by  «*+  l\ 

23.  r'  -  r  4-  1  by  r'  +  r  +  1. 

24.  m^  +  Sm'^i  +  wi/^'  by  W2  —  w. 


w 


Multiply  the  following : 

13.  a  ^l\>y  a  ^  b. 

14.  flj  —  J  by  a  —  ^. 
16.  a  -\-  b\iy  a  —  b. 

16.  a;  +  9  by  a;  -  9. 

17.  m  +  5  by  m  —  2. 

18.  Zy  +  7;2  by  4;?  +  by. 

25.  How  many  square  rods  in  a  field 
m  ■\-  n  rods  square  ? 

26.  How  many  acres  in  a  field  whose 
length  is  20/»  rods,  and  whose  width  is 
16m  rods  ? 

27.  A  has  hb  acres,  B  has  76'  acres, 
and  C  has  Via  acres.  They  offer  to 
sell  out  to  D  at  3w  dollars  an  acre. 
Find  what  D  would  have  to  pay. 

28.  A  man  having  "la  horses  sold  b  of  them  at  %r  each,  and 
the  remainder  at  %%r  each.  If  they  cost  him  %2>ar,  how  much 
did  he  gain  ? 

Find  the  value  of  the  following  : 

29.  {a  +  b)\  31.  {c  4-  3)'. 

30.  {a  -  b)\  32.  (y  +  2)  (?/  +  2). 


m« 

mn 

mn 

n^ 

436  SCHOOL  ARlfHMEf id. 

33.  (x'  +  7)  {x'  -  7).  35.  (m  -  7i)  (m  -  n), 

34.  {c  +  4.d)  {4d  +  c).  36.  {x  +  4:)  {x  +  5). 

37.  Show  that  the  square  of  the  sum  of  two  numbers  is 
equal  to  the  square  of  the  first  number,  plus  twice  the  prod- 
uct of  the  two  numbers,  plus  the  square  of  the  second 
number. 

38.  Square  a  -\-  b  and  a  —  h,  and  compare  the  results. 
The  square  of  the  difference  of  two  numlers  is  equal  to  what  ? 

39.  Multiply  x  +  yhy  x  —  y,  and  note  the  product.  The 
2)roduct  of  the  sum  and  difference  of  two  numbers  is  equal  to 
the  difference  of  what  ? 

Write  the  products  of  the  following : 

40.  {c  +  d)  {c  +  d).  43.  {Zy  -  5)  (3y  -  5). 

41.  (c  -¥  d)  (c-  d).  44.  (2a;  +  y)  (2a;  -  y). 

42.  (m  -n)  (m-  n).  45.  (2«  +  b)  {2a  +  b). 

46.  What  two  equal  factors  produce  a"^  +  2ax  +  x^  ? 

47.  What  two  equal  factors  give  the  product  ¥—  2by  +^''? 

48.  What  two  factors  produce  m^  —  n^  ?    x"^  —  y"^  "^ 

49.  How  many  square  feet  in  a  room  a  -\-  b  feet  long  and 
a  —  b  feet  wide  ?  How  many  square  yards  in  the  room,  if 
a  =  15  and  Z*  =r  12  ? 

50.  Find  the  volume  of  a  cube  whose  edge  \&  m  —  n  inches. 

DIVISION. 

618.  Division  is  the  inverse  of  multiplication.  In  the 
latter  two  factors  are  given,  the  product  required.  In  the 
former  the  product  (dividend)  and  one  factor  (divisor)  are 
given,  the  other  factor  (quotient)  required.  Hence  the  law 
of  signs  may  be  derived  from  that  in  multiplication,  as 
follows  : 

Since  +3  x  +5  =  +15, 

Since  +3  x  -5  =  -15, 

Since  ~3  x  +5  =  -15, 

Since  -3  x  -5  =  +15, 


15 -^ 

+  3  = 

+  5. 

15 -^ 

+3  = 

-5. 

15  -^ 

-3  =: 

+5. 

15 -r 

-3  = 

-5. 

INTRODUCTION  TO  ALGEBRA.  437 

k 

Or,  using  a  and  h  to  denote  anyjtwo  numbers,  we  have 
+«  X   +J  =:  +flf.^,    .-.  ^ah  -^  +«  =  +5. 
+a  X  -5  =  -ah,    . '.  -ab  -r-  +a  =  -b. 
-ax   +b  =  -ab,    .*.  -ab  -^  -a  =  +^. 
-a  X  -b  ~  ^ab,    .*.  ^ab  -^  -a  —  -b. 

That  is,  like  signs  of  dividend  and  divisor  give  a  positive 
quotient  ;  unlike  signs,  a  negative  quotient. 

Divide  the  folio-wing  : 

1.  Qab  by  2.  6.  a'^b  by  -ab. 

2.  %x  by  U.  7.  -2;.y'  by  -rcy. 

3.  16c  by  -4c.  8.  -a^b'^c  by  rt^>c. 

4.  -1%  by  2.  9.  20w'  by  -4m. 

5.  -12«m  by  -3w.  10.  IGaVy  by  Saa:^. 

11.  Divide  4a'a;  -  6a^/  +  2ac'  by  2a. 

2a  [  4a'a;   —  Ga^y'  +  2ac' 
2ax     —  Uy""     +       c' 

12.  Divide  x^  —  3x''y  +  3xy''  —  y'  by  x  —  y. 


x'-3x'y  4-  3xy'-y' 

X  - 

-y 

x\x-y)  =^x'-    x'y 

x'- 

-  'Zxy  +  y^. 

-  %x^y  H-  3a;^* 
-'Ixyix  —  y)  —  —  "Zx^y  +  2a:?/ 

xf  -  f 

f{^  -y)  =  ^f  -  f 

Divide  the  followinor : 

13.  a'  +  2ab  +  b^  hy  a  +  b. 

14.  m^  —  2mn  +  /^^  hj  m  —  n. 

15.  15^>^  -  8Jc  -  12c^  by  35  +2c. 

16.  x'  +  3a:>  +  ^xy""  -f  «/'  by  a:  +  ^y. 

17.  «/^  +3y*+2by«/  +  l. 

18.  a'  -  a  -  90  by  a  4-  9. 

19.  x^  —  y"^  hj  X  —  y. 

20.  9^^  -  4a^  bv  U  +  2a. 


4:38  School  arithmetic. 

21.  y"  -  \^yz  -  242;'  by  ^  +  %z. 

22.  16a;'  -  %^^xy  +  9«/'  by  4a:  -  3y. 

23.  There  are  16Jc  +  24c  square  feet  in  a  hall  8c  feet  widd. 
What  is  the  length  ? 

24.  The  area  of  a  field  «  +  c  rods  wide  is  a^  +  «6  +  3«c 
+  Jc  +  2c'  square  rods.     Find  the  length. 

Divide  the  following  by  the  highest  factor  common  to  all 
the  terms  : 

25.  %a^b  -  l^ahy.  27.  2c«/'  -  Qc'x  +  ^bc. 

26.  8Jaj'  -  6aZ»'a;.  28.  18m'  -  ^7nn  +  12m^i'. 
29.  Divide  «'  —  Z>'  first  by  a  +  ^  and  then  hy  a  —  h,  and 

compare  results.  The  difference  of  the  squares  of  two  num- 
bers is  divisible  by  what  ?  Then  what  are  the  two  factors  of 
a'  -  V  ? 

Write  from  inspection  the  factors  of  : 

30.  x^  -  y\  33.  4  -  h\ 

31.  x^  -  9.  34.  x^  -  a\j\ 

32.  a'  -  1.  35.  JV  -  d\ 

EQUATIONS. 

619.  An  equation  has  been  defined  as  a  statement  that 
two  numbers  or  expressions  are  equal. 

The  principles  that  apply  to  the  transformation  and  solution  of  equa- 
tions, as  given  in  Arts.  256-262,  should  here  be  reviewed. 

020.  In  solving  simple  integral  equations  the  following 
direction  will  be  found  useful  : 

Transpose  all  the  terms  containing  the  unknown  number  to 
the  first  member,  and  all  other  terms  to  the  second  member. 
Unite  like  terms,  arid  divide  both  members  by  the  coefficient 
of  the  unknown  number, 

621.  If  the  value  found  for  the  unknown  number  is  sub- 
stituted in  tbe  original  equation,  and  the  equation  reduces 
to  an  identity,  the  value  of  the  unknown  number  (called  the 
root  of  the  equation)  is  said  to  be  verified. 


INTRODUCTION  TO  ALGEBRA.  43^ 

Find  the  value  of  x,  and  verify  the  answer  : 

1.  5a;  =  28  -  2x.  4.  4a;  -  14  =  a;  -  2. 

2.  _  3a;  -  7  =  -  4.T  +  7.  5.  Hx  —  (ox  +  5)  =  7. 

3.  4ic  —  2(2  —  x)  =  Q.  6.  ax  =  mx  —  n. 

XX  X 

7.  Solve  the  equation  4  +  j=—  —  S+.y. 

X        X         ^        X 

Clearing  of  fractions,  48  +  3a;  =  6a;  —  3G  +  ^x. 

Uniting  terms,  —  7a;  =  —84. 

Dividing  by  —  7,  x  =  12. 

An  equation  may  be  cleared  of  fnictions  by  multiplying  both  mem- 
bers by  the  least  common  denominator  of  the  fractions,  which  in  this 
example  is  12, 

Solve  the  following : 

S.  x  +^=15.  12.  2/  +  I  +  f  :=  11. 

4  -'2       3 

9.. +1  =  18.  I3.i  +  i  +  |  =  l8. 

10.2«  +  i=U.  144^-1  +  1  =  8-^. 

11.  0J+*       j+f.  15.  l±I-l^^  ^UzA  =  I. 

4  4  4  o  2 

16.  Find  the  value  of  x  in  the  equation  a;"  —  4  =  5. 
x'  -4:  =  b. 

cc'  =  5  +  4  =  9. 

X  =  ±3,  by  extracting  the  square  root.     (Art.  521.) 
The  sign  ±  before  the  3,  read  plus  or  tninus,  shows  that  the  root  is 
either  +  or  — .      For  +3  x   +3  —  9,  and  -3  x  -3  =  9.      The  negative 
value  does  not  always  have  a  meaning  in  particular  problems. 

Solve  the  following  : 

17.  5x'  =  80.  20.  3a;'  +  1  =  2x''  +  10. 

18.  ^  -  5^5  =  ^'  -  4|.  21.  (x  +  Q)  {x-(j)  =  28. 

19.  (3  -  xy  =  3(1  -  xy.       22.  ax'  +  i  =  bx'  +  a. 


440  SCHOOL  ARITHMETIC. 


PROBLEMS. 

622.  In  stating  problems,  it  is  important  to  remember 
that  the  letter  x  should  not  be  put  for  distance,  time, 
weight,  etc.,  but  for  the  number  of  miles,  of  hours,  of 
pounds,  etc. 

In  connection  with  the  stating  and  solving  of  problems,  Arts.  261  and 
262  should  be  re-read. 

1.  The  sum  of  the  two  digits  of  a  number  is  4.  If  the 
digits  are  interchanged,  the  resulting  number  is  equal  to  the 
original  one.     What  is  the  number  ? 

Let  X  stand  for  the  digit  in  the  ones'  place. 
Then  4  —  a;  is  the  digit  in  the  tens'  place. 
.'.  10(4  —  ic)  +  a;  =  the  original  number.     Why  ? 
lOa^  +  (4  —  a;)  =  the  second  number. 
Hence,  10a;  -|-  (4  —  a;)  =  10(4  —  :iS)  -^^  x,  the  equation  of  the  problem. 
Solving  this  equation,  we  obtain 

a;  =  2,  the  digit  in  the  ones'  place  ; 
whence  4  —  a:  =  2,  the  digit  in  the  tens'  place. 
.-.  the  original  number  is  10(4  —  x)  +  x,  or  22. 

2.  A  son  is  one  fourth  as  old  as  his  father.  Four  years 
ago  he  was  only  one  fifth  as  old  as  his  father.  What  is  the 
age  of  each  ? 

Let        X  =  the  numher  of  years  in  the  father's  age. 

Then    —  =  the  number  of  years  in  the  son's  age.  . 
4 

a:  —  4  =  the  number  of  years  in  the  father's  age  4  years  ago. 

^  —  4  =  the  number  of  vears  in  the  son's  age  4  years  ago. 
4 

.-.  ^  —  4  =  i(a:  —  4),  whence 

X  =  64,  and^  =  16. 
4 

Therefore,  the  father  is  54  years  old  and  the  son  is  16  years  old. 

3.  A  certain  street  contains  144  square  rods,  and  the  length 
is  16  times  the  width.     Find  the  width. 


INTRODUCTION  TO  ALGEBRA.  441 

Let        X  =  the  number  of  rods  in  the  width  of  the  street. 
Then  16a;  =  the  number  of  rods  in  the  length  of  the  street. 
X  X  16a;   =  tlie  area  of  the  street  in  square  rods. 
16a;'  =  144 
a;"  =  9 
X   =  ±3 
Therefore,  the  width  of  the  street  is  3  rods.     The  negative  root  is  not 
applicable  to  this  particular  problem. 

4.  A  fulcrum  is  to  be  placed  under  a  3-foot  lever  so  as  to 
divide  it  into  two  parts  such  that  1.2  times  the  first  shall 
equal  4.8  times  the  second.  How  far  is  it  from  either 
end  ? 

5.  A  man  bought  10  yards  of  calico  and  20  yards  of  silk 
for  $30.60.  The  silk  cost  as  many  quarters  a  yard  as  the 
calico  cost  cents  a  yard.     Find  the  price  of  each. 

6.  I  bought  a  number  of  apples  at  the  rate  of  3  for  a  cent ; 
sold  one  third  of  them  at  2  for  a  cent,  and  the  remainder  at 
5  for  3  cents,  gaining  7  cents.     How  many  did  I  buy  ? 

7.  Atmospheric  air  is  a  mixture  of  four  parts  of  nitrogen 
with  one  of  oxygen.  How  many  cubic  feet  of  oxygen  are 
there  in  a  room  10  yd.  long,  5  yd.  wide,  and  12  ft.  high  ? 

8.  In  a  certain  family  each  son  has  as  many  brothers  as 
sisters,  but  each  daughter  has  twice  as  many  brothers  as  sis- 
ters.    How  many  children  are  in  the  family  ? 

9.  A  number  is  composed  of  two  digits  whose  sum  is  8. 
If  the  digits  are  interchanged,  the  resulting  number  is  greater 
by  18  than  the  original  number.     What  is  the  number  ? 

10.  One  third  of  my  sheep  equals  one  ninth  of  them  plus 
8.     How  many  have  I  ? 

11.  A  man  spends  ^of  his  income  for  rent,  ^  for  groceries, 
and  has  $1140  left.     What  is  his  income  ? 

12.  Divide  15  apples  between  A  and  B  so  that  ^  of  A's 
number  shall  equal  -J  of  B's. 

13.  My  wife's  age  plus  mine  equals  76  years,  and  f  of  her 
age  minus  two  years  equals  ^  of  my  age  plus  2  years.  Find 
the  age  of  each. 


442  SCHOOL  ARITHMETIC. 

14.  A  has  8150  more  than  B,  and  0  has  ^  as  much  as  A 
and  B.     They  all  have  $1000.     How  much  has  each  ? 

15.  Sixty  dollars  was  divided  equally  among  a  number  of 
men.  Had  their  number  been  4  less,  each  would  have  re- 
ceived three  times  as  much.     How  many  men  were  there  ? 

16.  Find  two  consecutive  numbers  such  that  J  of  the 
greater  is  3  more  than  \  of  the  less. 

17.  What  number  added  to  the  numerator  and  denomina- 
tor of  f  will  give  a  fraction  equal  to  |  ? 

18.  Eleven  sixteenths  of  a  certain  principal  was  at  interest 
at  5  per  cent,  and  the  remainder  at  4  per  cent.  The  entire 
income  was  11500.     Find  the  principal. 

19.  Two  numbers  are  to  each  other  as  3  to  4.  If  10  is  sub- 
tracted from  each,  the  smaller  one  will  be  f  of  the  larger. 
What  are  the  numbers  ? 

20.  Two  numbers  are  to  each  other  as  2  to  3,  and  their 
product  is  150.     What  are  the  numbers  ? 

21.  A  rectangular  lot  contains  an  acre,  and  its  width  is  |- 
of  its  length.     What  is  its  width  ? 

22.  A  triangular  field  contains  5  acres,  and  its  altitude  is 
■f  of  its  base.     What  is  the  base  ? 

23.  A  circular  pond  contains  314.16  square  yards.  AVhat 
is  its  diameter  ? 

24.  The  area  of  one  square  field  is  twice  that  of  another, 
and  they  together  contain  867  square  rods.  Find  the  length 
of  a  side  of  the  smaller. 

25.  A  rectangle  has  its  length  6  feet  longer  and  its  width 
5  feet  shorter  than  the  side  of  an  equivalent  square,  'Find 
its  area. 


THIS  BOOK  IS  DUU  ON  THF.  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL   BE  ASSESSED    FOR    FAILURE  TO    RETURN 
THIS    BOOK   ON    THE   DATE   DUE.    THE   PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY    AND    TO     $1.00    ON     THE    SEVENTH     DAY 
OVERDUE. 

■    JUL  27  193b 

LD  21-100m-8,'34 

I     I  v_y<^i^ 


PRICE-LIST  OF  TEXT  BOOKS. 


1.  Johnson's  Primer  (in  colors),  Paper  Biurltng, 

2.  Johnson's  Primer  (in  colors),  Bound  in  Boards 

3.  Johnson's  First  Reader,  Bound  in  Boards  

4.  Johnson's  First  Reader,  Bound  in  Cloth 

5.  Johnson's  Second  Header,  Hound  in  Boards 

(>.    Johnson's  Second  Reader,  Bound  in  Cloth 

7.  Johnson's  Third  Reader,  Bound  in  Cloth 

8.  Johnson's  Fourth  Reader,  Bound  in  Cloth 

a.     Johnson's  Fifth  Reader,  Bound  in  Cloth 

10.  Branson's  Common  School  Speller,  First  Hook  (In  Boards) 

11.  Branson's  Common  School  Speller,  Second  Book  (In  Boards)  

12.  Branson's  Common  School  Speller,  Complete  (In  Cloth) 

1.3.  Lee's  New  Prlmarj    History  of  the  United  States  (formerly  Lce'i 

Primary)  

14.  Lee's  New  School  History  of  the  United  States,  Cloth 

15.  Lee's  New  School  History  of  the  United  Stales,  Half  MortKCo 

16.  Lee's  Advanced  School   History   of  the   United   States 

17.  Smithdeal's  Slant  Writing- Books,  (six  numbers),  each 

18.  Johnson's  Vertical  Writing-Books  (Nog.  1, 2,  3,  4),  eacb... 
!»»'-  '                       •♦ical  WriiiiDi-,BoDkiaJina-&-»«va-iU 


911275 


THE  UNIVERSITY  OF  CALIFORNIA  UBRARY 


5^.    Southern  Literature  (from  1579  to  1895) 

53.  History  of  Virginia  (.Maury's) 

54.  Geography  of  Virginia  (Henning's) 

55.  Moses'  First  Phonetic  Reader,  by  Prof.  E.  P.  Moses,  B&leigh,  N.  C. 

(Other  numbers  in  course  of  preparation) 

56.  Some  Birds  and  Their  Ways,  cloth 

57.  Grimm's  Fairy  Tales,  by  P.P.  Claxton  and  Winifred  Haliburton,  Cloth 


For  Examination,  with  compliments  of 

B,  F.  JOHNSON  PUBLISHING  CO.,  Publishers, 

RicHt-t-iOttdl,  Vingit-ifa. 


I 


